Q  A 
685 
B62 
1896 

MAIN 


UC-NRLF 


B   M   EMfl 


THE  SCIENCE  ABSOLUTE  OF  SPACE 


Independent  of  the  Truth  or  Falsity  of  Euclid's 

Axiom  XI  (which  can  never  be 

decided  a  priori). 


BY 

JOHN    BOLYAI 
M 


TRANSLATED  FROM  THE  LATIN 


BY 
DR.  GEORGE  BRUCE  HALSTED 

PRESIDENT  OF  THE  TEXAS  ACADEMY  OF  SCIENCE 


FOURTH  EDITION. 


VOLUME    THREE   OF   THE   NEOMONIC   SERIES 


PUBLISHED  AT 

THE  XEOMON 

2407  Guadalupe  Street 
AUSTIN.  TEXAS,  U.  S.  A. 

1896 


TEANSLATOR'S  INTRODUCTION. 


The  immortal  Elements  of  Euclid  was  al- 
ready in  dim  antiquity  a  classic,  regarded  as 
absolutely  perfect,  valid  without  restriction. 

Elementary  geometry  was  for  two  thousand 
years  as  stationary,  as  fixed,  as  peculiarly 
Greek,  as  the  Parthenon.  On  this  foundation 
pure  science  rose  in  Archimedes,  in  Apollon- 
ius,  in  Pappus;  struggled  in  Theon,  in  Hypa- 
tia;  declined  in  Proclus;  fell  into  the  long 
decadence  of  the  Dark  Ages. 

The  book  that  monkish  Europe  could  no 
longer  understand  was  then  taught  in  Arabic 
by  Saracen  and  Moor  in  the  Universities  of 
Bagdad  and  Cordova. 

To  bring  the  light,  after  weary,  stupid  cen- 
turies, to  western  Christendom,  an  English- 
man, Adelhard  of  Bath,  journeys,  to  learn 
Arabic,  through  Asia  Minor,  through  Egypt, 
back  to  Spain.  Disguised  as  a  Mohammedan 
student,  he  got  into  Cordova  about  1120,  ob- 
tained a  Moorish  copy  of  Euclid's  Elements, 
and  made  a  translation  from  the  Arabic  into 
Latin. 


M306H52 


iv        TRANSLATOR'S  INTRODUCTION. 

The  first  printed  edition  of  Euclid,  pub- 
lished in  Venice  in  1482,  was  a  Latin  version 
from  the  Arabic.  The  translation  into  Latin 
from  the  Greek,  made  by  Zaniberti  from  a 
MS.  of  Theon's  revision,  was  first  published 
at  Venice  in  1505. 

Twenty-eight  years  later  appeared  the 
editio  princeps  in  Greek,  published  at  Basle 
in  1533  by  John  Hervagius,  edited  by  Simon 
Grynaeus.  This  was  for  a  century  and  three- 
quarters  the  only  printed  Greek  text  of  all  the 
books,  and  from  it  the  first  English  transla- 
tion (1570)  was  made  by  "Henricus  Billings- 
ley,"  afterward  Sir  Henry  Billingsley,  Lord 
Mayor  of  London  in  1591. 

And  even  to-day,  1895,  in  the  vast  system  of 
examinations  carried  out  by  the  British  Gov- 
ernment, by  Oxford,  and  by  Cambridge,  no 
proof  of  a  theorem  in  geometry  will  be  ac- 
cepted which  infringes  Euclid's  sequence  of 
propositions. 

Nor  is  the  work  unworthy  of  this  extraor- 
dinary immortality. 

Says  Clifford:  "This  book  has  been  for 
nearly  twenty-two  centuries  the  encourage- 
ment and  guide  of  that  scientific  thought 
which  is  one  thing  with  the  progress  of  man 
from  a  worse  to  a  better  state. 


TRANSLATOR'S  INTRODUCTION.          v 

"The  encouragement;  for  it  contained  a 
body  of  knowledge  that  was  really  known  and 
could  be  relied  on. 

"The  guide;  for  the  aim  of  every  student 
of  every  subject  was  to  bring  his  knowledge 
of  that  subject  into  a  form  as  perfect  as  that 
which  geometry  had  attained." 

But  Euclid  stated  his  assumptions  w4th  the 
most  painstaking  candor,  and  would  have 
smiled  at  the  suggestion  that  he  claimed  for 
his  conclusions  any  other  truth  than  perfect 
deduction  from  assumed  hypotheses.  In  favor 
of  the  external  reality  or  truth  of  those  as- 
sumptions he  said  no  word. 

Among  Euclid's  assumptions  is  one  differing 
from  the  others  in  prolixity,  whose  place  fluc- 
tuates in  the  manuscripts. 

Peyrard,  on  the  authority  of  the  Vatican  MS., 
puts  it  among  the  postulates,  and  it  is  often 
called  the  parallel-postulate.  Heiberg,  whose 
edition  of  the  text  is  the  latest  and  best  (Leip- 
zig, 1883-1888),  gives  it  as  the  fifth  postulate. 

James  Williamson,  who  published  the  closest 
translation  of  Euclid  we  have  in  English,  in- 
dicating, by  the  use  of  italics,  the  words  not 
in  the  original,  gives  this  assumption  as  elev- 
enth among  the  Common  Notions. 


vi        TRANSLATOR'S  INTRODUCTION. 

Bolyai  speaks  of  it  as  Euclid's  Axiom  XI. 
Todhunter  has  it  as  twelfth  of  the  Axioms. 

Clavius  (1574)  gives  it  as  Axiom  13. 

The  Harpur  Euclid  separates  it  by  forty- 
eight  pages  from  the  other  axioms. 

It  is  not  used  in  the  first  twenty-eight  pro- 
positions of  Euclid.  Moreover,  when  at  length 
used,  it  appears  as  the  inverse  of  a  proposition 
already  demonstrated,  the  seventeenth,  and  is 
only  needed  to  prove  the  inverse  of  another 
proposition  already  demonstrated,  the  twenty- 
seventh. 

Now  the  great  Lambert  expressly  says  that 
Proklus  demanded  a  proof  of  this  assumption 
because  when  inverted  it  is  demonstrable. 

All  this  suggested,  at  Europe's  renaissance, 
not  a  doubt  of  the  necessary  external  reality 
and  exact  applicability  of  the  assumption,  but 
the  possibility  of  deducing  it  from  the  other 
assumptions  and  the  twenty-eight  propositions 
already  proved  by  Euclid  without  it. 

Euclid  demonstrated  things  more  axiomatic 
by  far.  He  proves  what  every  dog  knows, 
that  any  two  sides  of  a  triangle  are  together 
greater  than  the  third. 

Yet  after  he  has  finished  his  demonstration, 
that  straight  lines  making  with  a  transversal 
equal  alternate  angles  are  parallel,  in  order  to 


TRANSLATOR'S  INTRODUCTION.        vii 

prove  the  inverse,  that  parallels  cut  by  a  trans- 
versal make  equal  alternate  angles,  he  brings 
in  the  unwieldy  assumption  thus  translated  by 
Williamson  (Oxford,  1781) : 

"11.  And  if  a  straight  line  meeting  two 
straight  lines  make  those  angles  which  are  in- 
ward and  upon  the  same  side  of  it  less  than 
two  right  angles,  the  two  straight  lines  being 
produced  indefinitely  will  meet  each  other  on 
the  side  where  the  angles  are  less  than  two 
right  angles." 

As  Staeckel  says,  "it  requires  a  certain 
courage  to  declare  such  a  requirement,  along- 
side the  other  exceedingly  simple  assumptions 
and  postulates."  But  was  courage  likely  to 
fail  the  man  who,  asked  by  King  Ptolemy  if 
there  were  no  shorter  road  in  things  geometric 
than  through  his  Elements?  answered,  "To 
geometry  there  is  no  special  way  for  kings!" 

In  the  brilliant  new  light  given  by  Bolyai 
and  Lobachevski  we  now  see  that  Euclid  un- 
derstood the  crucial  character  of  the  question 
of  parallels. 

There  are  now  for  us  no  better  proofs  of  the 
depth  and  systematic  coherence  of  Euclid's 
masterpiece  than  the  very  things  which,  their 
cause  unappreciated,  seemed  the  most  notice- 
able blots  on  his  work. 


viii      TRANSLATOR'S  INTRODUCTION. 

Sir  Henry  Savile,  in  his  Praelectiones  on 
Euclid,  Oxford,  1621,  p.  140,  says:  "In  pul- 
cherrimo  Geometriae  corpora  duo  sunt  naevi, 
duae  labes  ..."  etc.,  and  these  two  blemishes 
are  the  theory  of  parallels  and  the  doctrine  of 
proportion;  the  very  points  in  the  Elements 
which  now  arouse  our  wondering  admiration. 
But  down  to  our  very  nineteenth  century  an 
ever  renewing  stream  of  mathematicians  tried 
to  wash  away  the  first  of  these  supposed  stains 
from  the  most  beauteous  body  of  Geometry. 

The  year  1799  finds  two  extraordinary  young 
men  striving  thus 

4 '  To  gild  refined  gold,  to  paint  the  lily, 
To  cast  a  perfume  o'er  the  violet." 

At  the  end  of  that  year  Gauss  from  Braun- 
schweig writes  to  Bolyai  Farkas  in  Klausen- 
burg  (Kolozsvar)  as  follows:  [Abhandlungen 
der  Koeniglichen  Gesellschaft  der  Wissen- 
schaften  zu  Goettingen,  Bd.  22,  1877.] 

" 1  very  much  regret,  that  I  did  not  make  use 
of  our  former  proximity,  to  find  out  more 
about  your  investigations  in  regard  to  the  first 
grounds  of  geometry;  I  should  certainly  thereby 
have  spared  myself  much  vain  labor,  and  would 
have  become  more  restful  than  any  one,  such 


TRANSLATOR'S  INTRODUCTION.         ix 

as  I,  can  be,  so  long  as  on  such  a  subject  there 
yet  remains  so  much  to  be  wished  for. 

In  my  own  work  thereon  I  myself  have  ad- 
vanced far  (though  my  other  wholly  hetero- 
geneous employments  leave  me  little  time 
therefor)  but  the  way,  which  I  have  hit  upon, 
leads  not  so  much  to  the  goal,  which  one 
wishes,  as  much  more  to  making  doubtful  the 
truth  of  geometry. 

Indeed  I  have  come  upon  much,  which  with 
most  no  doubt  would  pass  for  a  proof,  but 
which  in  my  eyes  proves  as  good  as  nothing. 

For  example,  if  one  could  prove,  that  a  rec- 
tilineal triangle  is  possible,  whose  content  may 
be  greater,  than  any  given  surface,  then  I  am 
in  condition,  to  prove  with  perfect  rigor  all 
geometry. 

Most  would  indeed  l,et  that  pass  as  an  axiom; 
I  not;  it  might  well  be  possible,  that,  how  far 
apart  soever  one  took  the  three  vertices  of  the 
triangle  in  space,  yet  the  content  was  always 
under  a  given  limit. 

I  have  more  such  theorems,  but  in  none  do  I 
find  anything  satisfying." 

From  this  letter  we  clearly  see  that  in  1799 
Gauss  was  still  trying  to  prove  that  Euclid's 
is  the  only  non-contradictory  system  of  geome- 


x         TRANSLATOR'S  INTRODUCTION. 

try,  and  that  it  is  the  system  regnant  in  the 
external  space  of  our  physical  experience. 

The  first  is  false;  the  second  can  never  be 
proven. 

Before  another  quarter  of  a  century,  Bolyai 
Janos,  then  unborn,  had  created  another  pos- 
sible universe;  and,  strangely  enough,  though 
nothing  renders  it  impossible  that  the  space  of 
our  physical  experience  may,  this  very  year, 
be  satisfactorily  shown  to  belong  to  Bolyai 
Janos,  yet  the  same  is  not  true  for  Euclid. 

To  decide  our  space  is  Bolyai's,  one  need 
only  show  a  single  rectilineal  triangle  whose 
angle-sum  measures  less  than  a  straight  angle. 
And  this  could  be  shown  to  exist  by  imperfect 
measurements,  such  as  human  measurements 
must  always  be.  For  example,  if  our  instru- 
ments for  angular  measurement  could  be 
brought  to  measure  an  angle  to  within  one 
millionth  of  a  second,  then  if  the  lack  were  as 
great  as  two  millionths  of  a  second,  we  could 
make  certain  its  existence. 

But  to  prove  Euclid's  system,  we  must  show 
that  a  triangle's  angle-sum  is  exactly  a  straight 
angle,  which  nothing  human  can  ever  do. 

However  this  is  anticipating,  for  in  1799  it 
seems  that  the  mind  of  the  elder  Bolyai,  Bolyai 
Farkas,  was  in  precisely  the  same  state  as 


TRANSLATOR'S  INTRODUCTION.         xi 

that  of  his  friend  Gauss.  Both  were  intensely 
trying  to  prove  what  now  we  know  is  inde- 
monstrable. And  perhaps  Bolyai  got  nearer 
than  Gauss  to  the  unattainable.  In  his  * 4  Kurzer 
Grundriss  eines  Versuchs,"  etc.,  p.  46,  we  read: 
"Koennten  jede  3  Punkte,  die  nicht  in  einer 
Geraden  sind,  in  eine  Sphaere  fallen,  so  waere 
das  Eucl.  Ax.  XI.  bewiesen."  Frischauf  calls 
this  "das  anschaulichste  Axiom."  But  in  his 
Autobiography  written  in  Magyar,  of  which 
my  Life  of  Bolyai  contains  the  first  transla- 
tion ever  made,  Bolyai  Farkas  says:  "Yet  I 
could  not  become  satisfied  with  my  different 
treatments  of  the  question  of  parallels,  which 
was  ascribable  to  the  long  discontinuance  of 
my  studies,  or  more  probably  it  was  due  to 
myself  that  I  drove  this  problem  to  the  point 
which  robbed  my  rest,  deprived  me  of  tran- 
quillity." 

It  is  wellnigh  certain  that  Euclid  tried  his 
own  calm,  immortal  genius,  and  the  genius  of 
his  race  for  perfection,  against  this  self-same 
question.  If  so,  the  benign  intellectual  pride 
of  the  founder  of  the  mathematical  school  of 
the  greatest  of  universities,  Alexandria,  would 
not  let  the  question  cloak  itself  in  the  obscuri- 
ties of  the  infinitely  great  or  the  infinitely 
small.  He  would  say  to  himself:  "Can  I  prove 


xii       TRANSLATOR'S  INTRODUCTION. 


this   plain,   straightforward,   simple  theorem: 
^rhose  straights  which  are  produced  indefin- 


itely from  less  than  two  right  angles  meet. 
[This  is  the  form  which  occurs  in  the  Greek 
of  Eu.1.29.] 

Let  us  not  underestimate  the  subtle  power 
of  that  old  Greek  mind.  We  can  produce  no 
Venus  of  Milo.  Euclid's  own  treatment  of 
proportion  is  found  as  flawless  in  the  chapter 
which  StoU  devotes  to  it  in  1885  as  when 
through  Newton  it  first  gave  us  our  present 
continuous  number-system. 

But  what  fortune  had  this  genius  in  the  fight 
with  its  self-chosen  simple  theorem?  Was  it 
found  to  be  deducible  from  all  the  definitions, 
and  the  nine  "Common  Notions/'  and  the  five 
other  Postulates  of  the  immortal  Elements? 
Not  so.  But  meantime  Euclid  went  ahead 
without  it  through  twenty-eight  propositions, 
more  than  half  his  first  book.  But  at  last 
came  the  practical  pinch,  then  as  now  the  tri- 
angle's angle-sum. 

He  gets  it  by  his  twenty-ninth  theorem:  "A 
straight  falling  upon  two  parallel  straights 
makes  the  alternate  angles  equal." 

But  for  the  proof  of  this  he  needs  that  re- 
calcitrant proposition  which  has  how  long 
been  keeping  him  awake  nights  and  waking 


TRANSLATOR'S  INTRODUCTION.      xiii 

him  up  mornings?  Now  at  last,  true  man  of 
science,  he  acknowledges  it  indemonstrable  by 
spreading  it  in  all  its  ugly  length  among  his 
postulates. 

Since  Schiaparelli  has  restored  the  astron- 
omical system  of  Eudoxus,  and  Hultsch  has 
published  the  writings  of  Autolycus,  we  see 
that  Euclid  knew  surface-spherics,  was  famil- 
iar with  triangles  whose  angle-sum  is  more 
than  a  straight  angle.  Did  he  ever  think  to 
carry  out  for  himself  the  beautiful  system  of 
geometry  which  comes  from  the  contradiction 
of  his  indemonstrable  postulate;  which  exists 
if  there  be  straights  produced  indefinitely  from 
less  than  two  right  angles  yet  nowhere  meet- 
ing; which  is  real  if  the  triangle's  angle-sum 
is  less  than  a  straight  angle? 

Of  how  naturally  the  three  systems  of  geom- 
etry flow  from  just  exactly  the  attempt  we 
suppose  Euclid  to  have  made,  the  attempt  to 
demonstrate  his  postulate  fifth,  we  have  a  most 
romantic  example  in  the  work  of  the  Italian 
priest,  Saccheri,  who  died  the  twenty-fifth  of 
October,  1733.  He  studied  Euclid  in  the  edi- 
tion of  Clavius,  where  the  fifth  postulate  is 
given  as  Axiom  13.  Saccheri  says  it  should 
not  be  called  an  axiom,  but  ought  to  be  dem- 
onstrated. He  tries  this  seemingly  simple 


xiv       TRANSLATOR'S  INTRODUCTION. 

task;  but  his  work  swells  to  a  quarto  book  of 
101  pages. 

Had  he  not  been  overawed  by  a  conviction 
of  the  absolute  necessity  of  Euclid's  system, 
he  might  have  anticipated  Bolyai  Janos,  who 
ninety  years  later  not  only  discovered  the  new 
world  of  mathematics  but  appreciated  the 
transcendent  import  of  his  discovery. 

Hitherto  what  was  known  of  the  Bolyais 
came  wholly  from  the  published  works  of  the 
father  Bolyai  Farkas,  and  from  a  brief  article 
by  Architect  Fr.  Schmidt  of  Budapest  "Aus 
dem  Leben  zweier  ungarischer  Mathematiker, 
Johann  und  Wolfgang  Bolyai  von  Bolya." 
Grunert's  Archiv,  Bd.  48,  1868,  p.  217. 

In  two  communications  sent  me  in  Septem- 
ber and  October  1895,  Herr  Schmidt  has  very 
kindly  and  graciously  put  at  my  disposal  the 
results  of  his  subsequent  researches,  which  I 
will  here  reproduce.  But  meantime  I  have 
from  entirely  another  source  come  most  unex- 
pectedly into  possession  of  original  documents 
so  extensive,  so  precious  that  I  have  determined 
to  issue  them  in  a  separate  volume  devoted 
wholly  to  the  life  of  the  Bolyais;  but  these  are 
not  used  in  the  sketch  here  given. 

Bolyai  Farkas  was  born  February  9th,  1775, 
at  Bolya,  in  that  part  of  Transylvania  (Er- 


TRANSLATOR'S  INTRODUCTION.   xv 

dely)  called  Szekelyfrld.  He  studied  first  at 
Enyed,  afterward  at  Klausenburg  (Kolozsvar), 
then  went  with  Baron  Simon  Kemeny  to  Jena 
and  afterward  to  Goettingen.  Here  he  met 
Gauss,  then  in  his  19th  year,  and  the  two 
formed  a  friendship  which  lasted  for  life. 

The  letters  of  Gauss  to  his  friend  were  sent 
by  Bolyai  in  1855  to  Professor  Sartorius  von 
Walterhausen,  then  working  on  his  biography 
of  Gauss.  Everyone  who  met  Bolyai  felt  that 
he  was  a  profound  thinker  and  a  beautiful 
character. 

Benzenberg  said  in  a  letter  written  in  1801 
that  Bolyai  was  one  of  the  most  extraordinary 
men  he  had  ever  known. 

He  returned  home  in  1"7%  and  in  January, 
1804,  was  made  professor  of  mathematics  in 
the  Reformed  College  of  Maros-Vasarhely. 
Here  for  47  years  of  active  teaching  he  had 
for  scholars  nearly  all  the  professors  and  no- 
bility of  the  next  generation  in  Erdely. 

Sylvester  has  said  that  mathematics  is  poesy. 

Bolyai' s  first  published  works  were  dramas. 

His  first  published  book  on  mathematics  was 
an  arithmetic: 

Az  arithmetica  eleje.  8vo.  i-xvi,  1-162  pp. 
The  copy  in  the  library  of  the  Reformed  Col- 
lege is  enriched  with  notes  by  Bolyai  Janos. 


xvi       TRANSLATOR'S  INTRODUCTION. 

Next  followed  his  chief  work,  to  which  he 
constantly  refers  in  his  later  writings.  It  is 
in  Latin,  two  volumes,  8vo,  with  title  as  fol- 
lows: 

TENTAMEN    |    JUVENTUTEM     STUDIOSAM    | 
IN    ELEMENTA  MATHESEOS  PURAE,   ELEMEN- 
TARIS     AC    |    SUBLIMIORIS,     METHODO     INTUI- 
TIVA,  EVIDENTIA —      QUE    HUIC   PROPRIA,  IN- 
TRODUCENDI.    | 

CUM  APPENDICE  TRIPLICI.  |  Auctore  Pro- 
fessore  Matheseos  et  Physices  Chemiaeque  | 
Publ.  Ordinario.  |  Tomus  Primus.  .|  Maros 
Vasarhelyini.  1832.  |  Typis  Collegii  Re- 
formatorum  per  JosEPHUM,  et  |  SlMEONEM 
KALI  de  felso  Vist.  |  At  the  back  of  the  title: 
Imprimatur.  |  M.  Vasarhelyini  Die  |  12  Octo- 
bris,  1829.  |  Paulus  Horvath  m.  p.  Abbas, 
Parochus  et  Censor  |  Librorum. 

Tomus  Secundus.  |  Maros  Vasarhelyini. 
1833.  | 

The  first  volume  contains: 

Preface  of  two  pages:     Lectori  salutem. 

A  folio  table:     Explicatio  signorum. 

Index  rerum  (I  —  XXXII).  Errata 
(XXXIII— XXXVII). 

Pro.  tyronibus  prima  vice  legentibus  no- 
tanda  sequentia  (XXXVIII — LJI). 

Err  ores  (LIU— LXVI). 


TRANSLATOR'S  INTRODUCTION,      xvii 

Scholion  (LXVII— LXXIV). 

Plurium  errorum  haud  animadversorum 
numerus  minuitur  (LXXV — LXXVI). 

Recensio  per  auctorem  ipsum  facta 
(LXXVII— LXXVIII) . 

Err  ores     recentius     detecti     (LXXV- 
XCVIII). 

Now  comes  the  body  of  the  text  (pages 
1—502). 

Then,  with  special  paging,  and  a  new  title 
page,  comes  the  immortal  Appendix,  here 
given  in  English. 

Professors  Staeckel  and  Engel  make  a  mis- 
take in  their  "  Parallellinien "  in  supposing 
that  this  Appendix  is  referred  to  in  the  title 
of  "  Tentamen."  On  page  241  they  quote  this 
title,  including  the  words  '  *  Cum  appendice 
triplici,"  and  say:  "In  dem  dritten  Anhange, 
der  nur  28  Seiten  umfasst,  hat  Johann  Bolyai 
seine  neue  Geometrie  entwickelt." 

It  is  not  a  third  Appendix,  nor  is  it  refer- 
red to  at  all  in  the  words  ' '  Cum  appendice 
triplici." 

These  words,  as  explained  in  a  prospectus 
in  the  Magyar  language,  issued  by  Bolyai 
Parkas,  asking  for  subscribers,  referred  to  a 
real  triple  Appendix,  which  appears,  as  it 


xviii     TRANSLATOR'S  INTRODUCTION. 

should,  at  the  end  of  the  book  Tomus  Secun- 
dus,  pp.  265-322. 

The  now  world  renowned  Appendix  by 
Bolyai  Janos  was  an  afterthought  of  the 
father,  who  prompted  the  son  not  "to  occupy 
himself  with  the  theory  of  parallels,"  as 
Staeckel  says,  but  to  translate  from  the  Ger- 
man into  Latin  a  condensation  of  his  treatise, 
of  which  the  principles  were  discovered  and 
properly  appreciated  in  1823,  and  which  w-as 
given  in  writing  to  Johann  Walter  von  Eck- 
wehr  in  1825. 

The  father,  without  waiting  for-  Vol.  II, 
inserted  this  Latin  translation,  with  separate 
paging  (1-26),  as  an  Appendix  to  his  Vol.  I, 
where,  counting  a  page  for  the  title  and  a 
page  "Explicatio  signorum,"  it  has  twenty- 
six  numbered  pages,  followed  by  two  unnum- 
bered pages  of  Errata. 

The  treatise  itself,  therefore,  contains  only 
twenty-four  pages  —  the  most  extraordinary 
two  dozen  pages  in  the  whole  history  of 
thought! 

Milton  received  but  a  paltry  £5  for  his 
Paradise  Lost;  but  it  was  at  least  plus  £5. 

Bolyai  Janos,  as  we  learn  from  Vol.  II,  p. 
384,  of  ef  Tentamen"  contributed  for  the 


TRANSLATOR'S  INTRODUCTION.       xix 

printing-  of  his  eternal  twenty-six  pages,  104 
florins  50  kreuzers. 

That  this  Appendix  was  finished  consider- 
ably before  the  Vol.  I,  which  it  follows,  is 
seen  from  the  references  in  the  text,  breath- 
ing a  just  admiration  for  the  Appendix  and 
the  genius  of  its  author. 

Thus  the  father  says,  p.  452:  Elegans  est 
conceptus  similiumy  quern  J.  B.  Appendicis 
Auctor  dedit.  Again,  p.  489:  Appendicis 
Auctor,  rem  acumine  singulari  aggressus,  Ge- 
ometriam  pro  omni  casu  absolute  veram  posuit; 
quamvis  e  magna  mole,  tantum  summe  neces- 
saria,  in  Appendice  hujus  tomi  exhibuerit, 
multis  (ut  tetraedri  resolutione  generali,  plu- 
ribusque  aliis  disquisitionibus  elegantibus) 
brevitatis  studio  omissis. 

And  the  volume  ends  as  follows,  p.  502:  Nee 
operae  pretium  est  plura  referre;  quum  res 
tota  exaltiori  contemplationis  puncto,  in  ima 
penetranti  oculo,  tractetur  in  Appendice  se- 
quente,  a  quovis  fideli  veritatis  purae  alumno 
diagna  legi. 

The  father  gives  a  brief  resume  of  the  re- 
sults of  his  own  determined,  life-long,  desper- 
ate efforts  to  do  that  at  which  Saccheri,  J.  H. 
Lambert,  Gauss  also  had  failed,  to  establish 
Euclid's  theory  of  parallels  a  priori. 


xx       TRANSLATOR'S  INTRODUCTION. 

He  says,  p.  490:  "Tentamina  idcirco  quae 
olim  feceram,  breviter  exponenda  veniunt;  ne 
saltern  alius  quis  operam  eandem  perdat."  He 
anticipates  J.  Delboeuf's  * '  Prolegomenes  phil- 
osophiques  de  la  geometric  et  solution  des 
postulats,"  with  the  full  consciousness  in 
addition  that  it  is  not  the  solution, — that  the 
final  solution  has  crowned  not  his  own  intense 
efforts,  but  the  genius  of  his  son. 

This  son's  Appendix  which  makes  all  pre- 
ceding space  only  a  special  case,  only  a  species 
under  a  genus,  and  so  requiring  a  descriptive 
adjective,  Euclidean,  this  wonderful  produc- 
tion of  pure  genius,  this  strange  Hungarian 
flower,  was  saved  for  the  world  after  more 
than  thirty-five  years  of  oblivion,  by  the  rare 
erudition  of  Professor  Richard  Baltzer  of 
Dresden,  afterward  professor  in  the  Univer- 
sity of  Giessen.  He  it  was  who  first  did  jus- 
tice publicly  to  the  works  of  I^obachevski 
and  Bolyai. 

Incited  by  Baltzer,  in  1866  J.  Holiel  issued 
a  French  translation  of  Lobachevski's  Theory 
of  Parallels,  and  in  a  note  to  his  Preface  says: 
"M.  Richard  Baltzer,  dans  la  seconde  edition 
de  ses  excellents  Elements  de  Geometric,  a,  le 
premier,  introduit  ces  notions  exactes  a  la 
place  qu'elles  doivent  occuper,':  Honor  to 


TRANSLATOR'S  INTRODUCTION.       xxi 

Baltzer!  But  alas!  father  and  son  were  al- 
ready in  their  graves! 

Fr.  Schmidt  in  the  article  cited  (1868)  says: 
' '  It  was  nearly  forty  years  before  these  pro- 
found views  were  rescued  from  oblivion,  and 
Dr.  R.  Baltzer,  of  Dresden,  has  acquired  im- 
perishable titles  to  the  gratitude  of  all  friends 
of  science  as  the  first  to  draw  attention  to  the 
works  of  Bolyai,  in  the  second  edition  of  his 
excellent  Elemente  der  Mathematik  (1866-67). 
Following  the  steps  of  Baltzer,  Professor 
Hoiiel,  of  Bordeaux,  in  a  brochure  entitled, 
Essai  critique  sur  les  principes  fondamentaux 
de  la  Geometric  elementaire,  has  given  ex- 
tracts from  Bolyai's  book,  which  will  help  in 
securing  for  these  new  ideas  the  justice  they 
merit." 

The  father  refers  to  the  son's  Appendix 
again  in  a  subsequent  book,  Urtan  elemei  kez- 
doknek  [Elements  of  the  science  of  space  for 
beginners]  (1850-51),  pp.  48.  In  the  College 
are-  preserved  three  sets  of  figures  for  this 
book,  two  by  the  author  and  one  by  his  grand- 
son, a  son  of  Janos. 

The  last  work  of  Bolyai  Farkas,  the  only 
one  composed  in  German,  is  entitled, 

Kurzer  Grundriss  eines  Versuchs 

I.    Die  Arithmetik,  durch  zvekmassig  kons- 


xxii     TRANSLATOR'S  INTRODUCTION. 

truirte  Begriffe,  von  eingebildeten  und  unend- 
lich-kleinen  Grossen  gereinigt,  anschaulich 
und  logisch-streng  darzustellen. 

II.  In  der  Geometrie,  die  Begriffe  der  ger- 
aden  Linie,  der  Ebene,  des  Winkels  allgemein, 
der  winkellosen  Formen,  und  der  Krummen, 
der  verschiedenen  Arten  der  Gleichheit  u.  d. 
gl.  nicht  nur  scharf  zu  bestimmen;  sondern 
auch  ihr  Seyn  im  Raume  zu  beweisen:  und  da 
die  Frage,  ob  zwey  von  der  dritten  geschnit- 
tene  Geraden,  wenn  die  summe  der  inneren 
Winkel  nicht  =  2R,  sich  schneiden  oder 
nicht?  neimand  auf  der  Erde  ohne  ein  Axiom 
(wie  Euklid  das  XI)  aufzustellen,  beantworten 
wird;  die  davon  unabhilngige  Geometrie  ab- 
zusondern;  und  eine  auf  die  Ja — Antwort, 
andere  auf  das  Nein  so  zu  bauen,  das  die 
Formeln  der  letzten,  auf  ein  Wink  auch  in  der 
ersten  g^ltig  seven. 

Nach  ein  lateinischen  Werke  von  1829,  M. 
Vasarhely,  und  eben  daselbst  gedruckten  un- 
grischen. 

Maros  Vasarhely  1851.     8vo.  pp.  88. 

In  this  book  he  says,  referring  to  his  son's 
Appendix:  "Some  copies  of  the  work  pub- 
lished here  were  sent  at  that  time  to  Vienna, 
to  Berlin,  to  Goettingen.  .  .  .  From  Goet- 
tingen  the  giant  of  mathematics,  who  from 


TRANSLATOR'S  INTRODUCTION,     xxiii 

pinnacle  embraces  in  the  same  view  the 
and  the  abysses,  wrote  that  he  was  sur- 
£rised  to  see  accomplished  what  he  had  be- 
gun, only  to  leave  it  behind  in  his  papers." 

This  refers  to  1832.  The  only  other  record 
that  Gauss  ever  mentioned  the  book  is  a  letter 
from  Gerling,  written  October  31st,  1851,  to 
Wolfgang  Boylai,  on  receipt  of  a  copy  of 
"Kurzer  Grundriss."  Gerling,  a  scholar  of 
Gauss,  had  been  from  1817  Professor  of  As- 
tronomy at  Marburg.  He  writes :  4 '  I  do  not 
mention  my  earlier  occupation  with  the  theory 
of  parallels,  for  already  in  the  year  1810-1812 
with  Gauss,  as  earlier  1809  with  J.  F.  Pfaff  I 
had  learned  to  perceive  how  all  previous  at- 
tempts to  prove  the  Euclidean  axiom  had  mis- 
carried. I  had  then  also  obtained  preliminary 
knowledge  of  your  works,  and  so,  when  I  first 
[1820]  had  to  print  something  of  my  view 
thereon,  I  wrote  it  exactly  as  it  yet  stands 
to  read  on  page  187  of  the  latest  edition. 

"We  had  about  this  time  [1819]  here  a  law 
professor,  Schweikart,  who  was  formerly  in 
Charkov,  and  had  attained  to  similar  ideas, 
since  without  help  of  the  Euclidean  axiom  he 
developed  in  its  beginnings  a  geometry  which 
he  called  Astralgeometry.  What  he  commun- 
icated to  me  thereon  I  sent  to  Gauss,  who 


xxiv     TRANSLATOR'S  INTRODUCTION. 

then  informed  me  how  much  farther  alre 
had  been  attained  on  this  way,  and  later 
expressed  himself  about  the  great  acquisitio 
which  is  offered  to  the  few  expert  judges  ii, 
the  Appendix  to  your  book." 

The  ' '  latest  edition ' '  mentioned  appeared 
in  1851,  and  the  passage  referred  to  is:  "This 
proof  [of  the  parallel-axiom]  has  been  sought 
in  manifold  ways  by  acute  mathematicians, 
but  yet  until  now  not  found  with  complete 
sufficiency.  So  long  as  it  fails,  the  theorem, 
as  all  founded  on  it,  remains  a  hypothesis, 
whose  validity  for  our  life  indeed  is  suffici- 
ently proven  by  experience,  whose  general, 
necessary  exactness,  however,  .could  be 
doubted  without  absurdity." 

Alas!  that  this  feeble  utterance  should  have 
seemed  sufficient  for  more  than  thirty  years 
to  the  associate  of  Gauss  and  Schweikart,  the 
latter  certainly  one  of  the  independent  discov- 
erers of  the  non-Euclidean  geometry.  But 
then,  since  neither  of  these  sufficiently  real- 
ized the  transcendent  importance  of  the  mat- 
ter to  publish  any  of  their  thoughts  on  the 
subject,  a  more  adequate  conception  of  the 
issues  at  stake  could  scarcely  be  expected  of 
the  scholar  and  colleague.  How  different  with 
Bolyai  Janos  and  Lobachevski,  who  claimed 


TRANSLATOR'S  INTRODUCTION.      xxv 

at  once,  unflinchingly,  that  their  discovery 
marked  an  epoch  in  human  thought  so  momen- 
tous as  to  be  unsurpassed  by  anything  re- 
corded in  the  history  of  philosophy  or  of 
science,  demonstrating  as  had  never  been 
proven  before  the  supremacy  of  pure  reason 
at  the  very  moment  of  overthrowing  what 
had  forever  seemed  its  surest  possession,  the 
axioms  of  geometry. 

On  the  9th  of  March,  1832,  Bolyai  Farkas 
was  made  corresponding  member  in  the  math- 
ematics section  of  the  Magyar  Academy. 

As  professor  he  exercised  a  powerful  in- 
fluence in  his  country. 

In  his  private  life  he  was  a  type  of  true 
originality.  He  wore  roomy  black  Hungarian 
pants,  a  white  flannel  jacket,  high  boots,  and 
a  broad  hat  like  an  old-time  planter's.  The 
smoke-stained  wall  of  his  antique  domicile 
was  adorned  by  pictures  of  his  friend  Gauss, 
of  Schiller,  and  of  Shakespeare,  whom  he 
loved  to  call  the  child  of  nature.  His  violin 
was  his  constant  solace. 

He  died  November  20th,  1856.  It  was  his 
wish  that  his  grave  should  bear  no  mark. 

The  mother  of  Bolyai  Janos,  nee  Arkosi 
Benka  Zsuzsanna,  was  beautiful,  fascinating, 


xxvi     TRANSLATOR'S  INTRODUCTION. 

of  extraordinary  mental  capacity,  but  always 
nervous. 

Janos,  a  lively,  spirited  boy,  was  taught 
mathematics  by  his  father.  His  progress  was 
marvelous.  He  required  no  explanation  of 
theorems  propounded,  and  made  his  own  dem- 
onstrations for  them,  always  wishing  his 
father  to  go  on.  "Like  a  demon,  he  always 
pushed  me  on  to  tell  him  more." 

At  12,  having  passed  the  six  classes  of  the 
Latin  school,  he  entered  the  philosophic-cur- 
riculum, which  he  passed  in  two  years  with 
great  distinction. 

When  about  13,  his  father,  prevented  from 
meeting  his  classes,  sent  his  son  in  his  stead. 
The  students  said  they  liked  the  lectures  of 
the  son  better  than  those  of  the  father.  He 
already  played  exceedingly  well  on  the  violin. 

In  his  fifteenth  year  he  went  to  Vienna  to 
K.  K.  Ingenieur-Akademie. 

In  August,  1823,  he  was  appointed  "sous- 
lieutenant"  and  sent  to  Temesvar,  where  he 
was  to  present  himself  on  the  2nd  of  Sep- 
tember. 

From  Temesvar,  on  November  3rd,  1823, 
Janos  wrote  to  his  father  a  letter  in  Magyar, 
of  which  a  French  translation  was  sent  me  by 
Professor  Koncz  Jozsef  on  February  14th, 


TRANSLATOR'S  INTRODUCTION,    xxvii 

1895.     This  will  be  given  in  full  in  my  life  of 
Bolyai;  but  here  an  extract  will  suffice: 

"My  Dear  and  Good  Father: 

"I  have  so  much  to  write  about  my  new 
inventions  that  it  is  impossible  for  the  mo- 
ment to  enter  into  great  details,  so  I  write 
you  only  on  one-fourth  of  a  sheet.  I  await 
your  answer  to  my  letter  of  two  sheets;  and 
perhaps  I  w^ould  not  have  written  you  before 
receiving  it,  if  1  had  not  wished  to  address  to 
you  the  letter  I  am  writing  to  the  Baroness, 
which  letter  I  pray  you  to  send  her. 

*  *  First  of  all  I  reply  to  you  in  regard  to  the 

binominal. 

•*##          #          *          #### 

"Now  to  something  else,  so  far  as  space 
permits.  I  intend  to  write,  as  soon  as  I  have 
put  it  into  order,  and  when  possible  to  pub- 
lish, a  work  on  parallels. 

"At  this  moment  it  is  not  yet  finished, 
but  the  way  which  I  have  followed  promises 
me  with  certainty  the  attainment  of  the  goal, 
if  it  in  general  is  attainable.  It  is  not  yet 
attained,  but  I  have  discovered  such  magnifi- 
cent things  that  I  am  myself  astonished  at 
them. 

"It  would  be  damage  eternal  if  they  were 


xxviii  TRANSLATOR'S  INTRODUCTION. 

lost.  When  you  see  them,  my  father,  you 
yourself  will  acknowledge  it.  Now  I  can  not 
say  more,  only  so  much:  that  from  nothing  I 
have  created  another  wholly  new  world.  All 
that  I  have  hitherto  sent  you  compares  to  this 
only  as  a  house  of  cards  to  a  castle. 

"P.  S. — I  dare  to  judge  absolutely  and  with 
conviction  of  these  works  of  my  spirit  before 
you,  my  father;  I  do  not  fear  from  you  any 
false  interpretation  (that  certainly  I  would 
not  merit),  which  signifies  that,  in  certain 
regards,  I  consider  you  as  a  second  self." 

From  the  Bolyai  MSS.,  now  the  property  of 
the  College  at  Maros-Vasarhely,  Fr.  Schmidt 
has  extracted  the  following  statement  by 
Janos : 

"First  in  the  year  1823  have  I  pierced 
through  the  problem  in  its  essence,  though 
also  afterwards  completions  yet  were  added. 

'  *  I  communicated  in  the  year  1825  to  my 
former  teacher,  Herr  Johann  Walter  von  Eck- 
wehr  (later  k.  k.  General)  [in  the  Austrian 
Army],  a  written  treatise,  which  is  still  in 
his  hands. 

4 '  On  the  prompting  of  my  father  I  trans- 
lated my  treatise  into  the  Latin  language,  and 


TRANSLATOR'S  INTRODUCTION,     xxix 

it  appeared  as  Appendix  to  the  Tentamen, 
1832." 

The  profound  mathematical  ability  of  Bol- 
yai  Janos  showed  itself  physically  not  only  in 
his  handling  of  the  violin,  where  he  was  a 
master,  but  also  of  arms,  where  he  was  unap- 
proachable. 

It  was  this  skill,  combined  with  his  haughty 
temper,  which  caused  his  being  retired  as  Cap- 
tain on  June  16th,  1833,  though  it  saved  him 
from  the  fate  of  a  kindred  spirit,  the  lamented 
Galois,  killed  in  a  duel  when  only  19.  Bolyai, 
when  in  garrison  with  cavalry  officers,  was 
provoked  by  thirteen  of  them  and  accepted  all 
their  challenges  on  condition  that  he  be  per- 
mitted after  each  duel  to  play  a  bit  on  his 
violin.  He  came  out  victor  from  his  thirteen 
duels,  leaving  his  thirteen  adversaries  on  the 
square. 

He  projected  a  universal  language  for 
speech  as  we  have  it  for  music  and  for  mathe- 
matics. 

He  left  parts  of  a  book  entitled:  Principia 
doctrinae  novae  quantitatum  imaginariarum 
perfectae  uniceque  satisfacientis,  aliaeque  dis- 
quisitiones  analyticae  et  analytico  -  geome- 
tricae  cardinales  gravissimaeque;  auctore 


TRANSLATOR'S  INTRODUCTION. 


Johan.  Bolyai  de  eadem,  C.  R.  austriaco  cas- 
trensium  captaneo  pensionato. 

Vindobonae  vel  Maros  Vasarhelyini,  1853. 

Bolyai  Farkas  was  a  student  at  Goettingen 
from  1796  to  1799. 

In  1799  he  returned  to  Kolozsvar,  where 
Bolyai  Janos  was  born  December  18th,  1802. 

He  died  January  27th,  1860,  four  years 
after  his  father. 

In  1894  a  monumental  stone  was  erected  on 
his  long-neglected  grave  in  Maros-Vasarhely 
by  the  Hungarian  Mathematico-Physical  So- 
ciety. 


APPENDIX. 

SCIENTIAM  SPATII  absolute  veram  exhibens: 

veritate  aut  falsitate  Axiomatis  XI  Euclidei 

(a  priori  haud  unquam  decidendd]   in- 

dependentem:  adjecta  ad  casum  fal- 

sitatis,    quadratura    circuli 

geometrica. 


Vuctore  JOHANNE  BOLYAI  de  eadem,  Geometrarum 

in  Exercitu  Caesareo  Regio  Austriaco 

Castrensium  Capitaneo. 


EXPLANATION  OF  SIGNS. 


The  straight  AB  means  the  aggregate  of  all  points  situated 
in  the  same  straight  line  with  A  and  B. 

The  sect  AB  means  that  piece  of  the  straight  AB  between 
the  points  A  and  B.  ^ 

The  ray  AB  means  that  half  of  the  straight  AB  which  com- 
mences at  the  point  A  and  contains  the  point  B. 

The  plane  ABC  means  the  aggregate  of  all  points  situated 
in  the  same  plane  as  the  three  points  (not  in  a 
straight)  A,  B,  C. 

The  hemi-plane  ABC  means  that  half  of  the  plane  ABC 
which  starts  from  the  straight  AB  and  contains  the 
point  C. 

ABC  means  the  smaller  of  the  pieces  into  which  the  plane 
ABC  is  parted  by  the  rays  BA,  BC,  or  the  non-reflex 
angle  of  which  the  sides  are  the  rays  BA,  BC. 

ABCD  (the  point  D  being  situated  within  /ABC,  and  the 
straights  BA,  CD  not  intersecting)  means  the  portion 
of  /  ABC  comprised  between  ray  BA,  sect  BC,  raj' 
CD;  while  BACD  designates  the  portion  of  the  plane 
ABC  comprised  between  the  straights  AB  and  CD. 

J_  is  the  sign  of  perpendicularity. 
||  is  the  sign  of  parallelism. 

/  means  angle. 

rt.  /  is  right  angle. 

st.  /  is  straight  angle. 

^  is  the  sign  of  congruence,  indicating  tnat  two  magni- 
tudes are  superposable. 

AB-CD  means  /CAB=/ACD. 

x=a  means  x  converges  toward  the  limit  a. 

A  is  triangle. 

Qr  means  the  [circumference  of  the]  circle  of  radius  r. 

area  Qr  means  the  area  of  the  surface  of  the  circle  of  radius  r. 


THE  SCIENCE  ABSOLUTE  OF  SPACE. 


the  ray  AM  is  not  cut  by  the  ray  [3] 
BN,  situated  in  the  same  plane,  but 
is  cut  by  every  ray  BP  comprised 
in  the  angle  ABN,  we  will  call  ray 
BN  parallel  to  ray  AM;  this  is 
designated  by  BN  II  AM. 

It  is  evident  that  there  is  one 
such  ray  BN ,  and  only  one,  pass- 
ing through  any  point  B  (taken  out- 
side of  the  straight  AM),  and  that 
the  sum  of  the  angles  BAM,  ABN 
exceed  a  st.  Z  ;  for  in  moving  BC 
around  B  until  BAM+ABC=st.  /,  somewhere 
ray  BC  first  does  not  cut  ray  AM,  and  it  is 
then  BCNAM.  It  is  clear  that  BN  II  EM, 
wherever  the  point  E  be  taken  on  the  straight 
AM  (supposing  in  all  such  cases  AM>AE). 

If  while  the  point  C  goes  away  to  infinity 
on  ray  AM,  always  CD=CB,  we  will  have  con- 
stantly CDB=(CBD<  NBC);  butNBC=0;  and 
so  also  ADB=0. 


FIG.  1. 
can  not 


SCIENCE  ABSOLUTE  OF  SPACE. 


FIG.  2. 


2.  If  BN  II  AM,  we  will  have  also  CN  II  AM. 
For  take  D  anywhere  in  MACN. 
If  C  is  on  ray  BN,  ray  BD  cuts 
ray  AM,  since  BN  II  AM,  and  so 
also  ray  CD  cuts  ray  AM.  But 
if  C  is  on  ray  BP,  take  BQ  II  CD; 
BQ  falls  within  the  Z  ABN  (§1), 
and  cuts  ray  AM;  and  so  also 
ray  CD  cuts  ray  AM.  Therefore 
every  ray  CD  (in  ACN)  cuts,  in 
peach  case,  the  ray  AM,  without 

CN  itself  cutting  ray  AM.     Therefore  always 

CN  II  AM. 

§  3.  (Fig.  2.)  If  BR  and  CS  and  each  II  AM, 
and  C  is  not  on  the  ray  BR,  then  ray  BR  and 
ray  CS  do  not  intersect.  For  if  ray  BR  and 
ray  CS  had  a  common  point  D,  then  (§  2)  DR 
and  DS  would  be  each  II  AM,  and  ray  DS  (§  1) 
would  fall  on  ray  DR,  and  C  on  the  ray  BR 
(contrary  to  the  hypothesis). 

4.    If  MAN>MAB,  we  will  have  for  every 
point  B  of  ray  AB,  a  point 
p  C   of   ray   AM,    such    that 
BCM=NAM. 

For   (by   §  1)    is    granted 
N   BDM>NAM,    and    so   that 
FIG.  3.  MDP-MAN,  and  B  falls  in 


SCIENCE  ABSOLUTE  OP  SPACE.          7 

NADP.  If  therefore  NAM  is  carried  along 
AM  until  ray  AN  arrives  on  ray  DP,  ray 
AN  will  somewhere  have  necessarily  passed 
through  B,  and  some  BCM=NAM. 

§  5.  If  BN  II  AM,  there  is  on  the  straight  w 
iN  AM  a  point  F  such  that  FM^BN. 
For  by  §  1  is  granted  BCM>CBN; 
and  if  CE-CB,  and  so  EC^BC; 
evidently  BEM<EBN.  The  point 
P  is  moved  on  EC,  the  angle  BPM 
always  being  called  u,  and  the  an- 
gle PBN  always  v;  evidently  u  is 
at  first  less  than  the  corresponding 
v,  but  afterwards  greater.  Indeed 
u  increases  continuously  from 
'•  4-  BEM  to  BCM;  since  (by  §  4)  there 
exists  no  angle  >BEM  and  <BCM,  to  which 
u  does  not  at  some  time  become  equal.  Like- 
wise v  decreases  continuously  •  from  EBN  to 
CBN.  There  is  therefore  on  EC  a  point  F 
such  that  BFM=FBN. 

§6.  If  BNIIAM  and  E  anywhere  in  the 
straight  AM,  and  G  in  the  straight  BN;  then 
GN  II  EM  and  EM  II  GN.  For  (by  §  1)  BN  II  EM, 
whence  (by  §  2)  GN  II  EM.  If  moreover  FM^ 
BN  (§5) ;"  then  MFBN^NBFM,  and  conse- 
quently (since  BN  II  FM)  also  FM  II  BN,  and 
(by  what  precedes)  EM  II  GN. 


8 


SCIENCE  ABSOLUTE  OF  SPACE. 


§  7. 


N      M 


A 

FIG.  5. 


If  BN  and  CP  are  each  II  AM,  and  C 
not  on  the  straight  BN;  also  BN  II  CP. 
For  the  rays  BN  and  CP  do  not  in- 
tersect (§3);  but  AM,  BN  and  CP 
either  are  or  are  not  in  the  same 
plane;  and  in  the  first  case,  AM  either 
is  or  is  not  within  BNCP. 

If  AM,  BN,  CP  are  complanar,  and 
AM  falls  within  BNCP;  then  every  ray  BQ 
(in  NBC)  cuts  the  ray  AM  in  some  point  D 
(since  BN  II  AM)  ;  moreover,  since  DM  II  CP 
(§  6),  the  ray  DQ  will  cut  the  ray  CP,  and  so 
BN  II  CP. 

But  if  BN  and  CP  are  on  the  same  side  of 
M  AM;  then  one  of  them,  for  example 
CP,  falls  between  the  two  other 
straights  BN,  AM:  but  every  ray  BQ 
(in  NBA)  cuts  the  ray  AM,  and  so 
also  the  straight  CP.  Therefore 
BN  II  CP. 

If  the  planes  MAB,  MAC  make 
an  angle;  then  CBN  and  ABN  have  in  com- 
mon nothing  but  the  ray  BN,  while  the  ray 
AM  (in  ABN)  and  the  ray  BN,  and  so  also 
NBC  and  the  ray  AM  have  nothing  in  com- 
mon. 

But  hemi-plane  BCD,   drawn  through   any 
ray  BD  (in  NBA),  cuts  the  ray  AM,  since  ray 


B      C       A 

FIG.  6. 


SCIENCE  ABSOLUTE  OF  SPACE.         9 

BQ  cuts  ray  AM  fas  BNHAM>. 
Therefore  in  revolving  the  hemi-plane 
BCD  around  BC  until  it  begins  to 
leave  the  ray  AM,  the  hemi-plane 
BCD  at  last  will  fall  upon  the  hemi- 
plane  BCN.  For  the  same  reason  this 
same  will  fall  upon  hemi-plane  BCP. 
FIG.  7.  Therefore  BN  falls  in  BCP.  More- 
over, if  BR  II  CP;  then  (because  also  AM  II  CP) 
by  like  reasoning,  BR  falls  in  BAM,  and  also 
(since  BRIICP>  in  BCP.  Therefore  the 
straight  BR,  being  common  to  the  two  planes 
MAB,  PCB,  of  course  is  the  straight  BN,  and 
hence  BN  II  CP.* 

If.  therefore  CP  II  AM,  and  B  exterior  to  the 
plane  CAM;  then  the   intersection  BN  of  the 
planes  BAM,  BCP  is  II  as  well  to  AM  as  to  CP. 
§  8.    If  BN  II  and  ===  CP  (or  more  briefly  BN 
II  =*CP>,  and  AM  (inNBCP)  bisects 
J_  the  sect  BC;  then  BN  II  AM. 

For  if  ray  BN  cut  ray  AM,  also 
ray  CP  would  cut  ray  AM  at  the 
same     point     (because    MABN= 
c  MACP) ,  and  this  would  be  common 
to  the  rays  BN,  CP  themselves,  al- 


*  The  third  case  being  put  before  the  other  two,  these  can  be 
demonstrated  together  with  more  brevity  and  elegance,  like  ease 
2  of  §  10.  [Author's  note.] 


io        SCIENCE  ABSOLUTE  OF  SPACE. 

though  BN  II  CP.  But  every  ray  BQ  (in  CBN) 
cuts  ray  CP;  and  so  ray  BQ  cuts  also  ray  AM. 
Consequently  BN  II  AN.  . 

§  9.    If  BN  II  AM,  and  MAPlMAB,  and  the 
Z,    which   NBD    makes   with 
NBA  (on  that  side  of  MABN, 
where  MAP  is)  is  <rt.Z;  then 
MAP  and  NBD  intersect. 
B      For   let    ZBAM=rt.Z,    and 
AClBN    (whether   or   not   C 
falls  on  B),   and    CElBN  (in 
FIG.  9.          NBD);    by    hypothesis   ZACE 

,  and  AF  (j_CE)  will  fall  in  ACE. 
Let  ray  AP  be  the  intersection  of  the  hemi- 
planes  ABF,  AMP  (which  have  the  point  A 
common);  since  BAM ± MAP,  ZBAP-ZBAM 
=  rt.Z. 

If  finally  the  hemi-plane  ABF  is  placed  upon 
the  hemi-plane  ABM  (A  and  B  remaining),  ray 
AP  will  fall  on  ray  AM;  and  since  AClBN, 
and  sect  AF<sect  AC,  evidently  sect  AF  will 
terminate  within  ray  BN,  and  so  BF  falls  in 
ABN.  But  in  this  position,  ray  BF  cuts  ray  AP 
(because  BN  II  AM) ;  and  so  ray  AP  and  ray  BF 
intersect  also  in  the  original  position;  and  the 
point  of  section  is  common  to  the  hemi-planes 
MAP  and  NBD.  Therefore  the  hemi-planes 
MAP  and  NBD  intersect.  Hence  follows  eas- 


SCIENCE  ABSOLUTE  OF  SPACE. 


11 


ily  that  the  hemi-planes  MAP  and  NBD  inter- 
sect if  the  sum  of  the  interior  angles  which 
they  make  with  MABN  is  <st.Z. 

§10.    If  both  BN  and  CPU  ^AM;    also  is 

BNII  ^CP. 

For  either  MAB 
and  MAC  make  an 
angle,  or  they  are  in 
a  plane. 

If  the  first;  let  the 
hemi-plane  QDF  bi- 
sect _L  sect  AB;  then 
DQlAB,  and  so  DQ 
II  AM  (§  8) ;  likewise  if  hemi-plane  ERS  bisects 
1  sect  AC,  is  ER  II  AM;  whence  (§  7)  DQ  II  ER. 
Hence  follows  easily  (by  §9),  the  hemi- 
planes  QDF  and  ERS  intersect,  and  have  (§  7) 
their  intersection  FS  II  DO,  and  (on  account  of 
BNIIDQ)  also  FS  II  BN.  Moreover  (for  any 
point  of  FS)  FB=FA=FC,  and  the  straight 
FS  falls  in  the  plane  TGF,  bisecting  J_  sect  BC. 
But  (by  §7)  (since  FS  II  BN)  also  GT  II  BN. 
In  the  same  way  is  proved  GT  II  CP.  Mean- 
while GT  bisects  1  sect  BC;  and  so  TGBN^ 
TGCP  (§1),  andBNll^CP. 

If  BN,  AM  and  CP  are  in  a  plane,  let  (fall- 
ing without  this  plane)  FS  II  —AM;  then  (from 


12        SCIENCE  ABSOLUTE  OF  SPACE. 

what  precedes)  FS  II  ===  both  to  BN  and  to  CP, 
and  so  also  BN  II  ^=CP. 

§  11.  Consider  the  aggregate  of  the  point 
A,  and  all  points  of  which  any  one  B  is  such, 
that  if  BN  II  AM,  also  BN^AM;  call  it  F;  but 
the  intersection  of  F  with  any  plane  contain- 
ing the  sect  AM  call  L. 

F  has  a  point,  and  one  only,  on  any  straight 
II  AM;  and  evidently  L  is  divided  by  ray  AM 
into  two  congruent  parts. 

Call  the  ray  AM  the  axis  of  L.  Evidently 
also,  in  any  plane  containing  the  sect  AM,  there 
is  for  the  axis  ray  AM  a  single  L.  Call  any 
L  of  this  sort  the  L  of  this  ray  AM  (in  the 
plane  considered,  being  understood).  Evi- 
dently by  revolving  L  around  AM  we  describe 
the  F  of  which  ray  AM  is  called  the  axis,  and  in 
turn  F  may  be  ascribed  to  the  axis  ro,y  AM. 

§  12.  If  B  is  anywhere  on  the  L  of  ray  AM, 
and  BN  II  ^  AM  (§  11) ;  then  the  L  of  ray  AM 
and  the  L  of  ray  BN  coincide.  For  suppose, 
in  distinction,  I/  the  L  of  ray  BN.  Let  C  be 
anywhere  in  I/,  and  CP  II  ^=BN  (§  11).  Since 
BN  II  ^AM,  so  CP  II  *=AM  (§  10),  and  so  C  also 
will  fall  on  L.  And  if  C  is  anywhere  on  L,  and 
CP  II  ^AM;  then  CP  II  ^BN  (§  10) ;  and  C  also 
falls  on  L'  (§11).  Thus  L  and  L'  are  the 


SCIENCE  ABSOLUTE  OF  SPACE. 


13 


same;  and  every  ray  BN  is  also  axis  of  L,  and 
between  all  axes  of  this  L,  is  —  . 

The  same  is  evident  in  the  same  way  of  F. 
§  13.    If  BN  II  AM,  and  CP  II  DQ,  andZBAM 
+ZABN=st.Z;    then   also    ZDCP+ZCDQ= 
st.Z. 

Q  For  let  EA= 
EB,  and  EFM= 
DCP  (§  4).  Since 
ZBAM+ZABN 


SN 


S  O 

I 


=st.  Z  = 
ZABG,   we  have 


FIG.  11. 


D  and  so  if  also  BG 
=AF,  thenAEBG 

\  ZBEG-ZAEF  and  G  will  fall  on 
the  ray  FE.    Moreover  ZGFM+ZFGN=st.  Z 
(since  ZEGB=ZEFA). 
AlsoGNllFM  (§6). 

Therefore  if  MFRS^PCDQ,  then  RSllGN 
(§  7),  and  R  falls  within  or  without  the  sect 
FG  (unless  sect  CD  =  sect  FG,  where  the  thing 
now  is  evident). 

I.  In  the  first  case  ZFRS  is  not  >(st.Z-Z 
RFM=ZFGN),  since  RSllFM.  But  as  RSl'l 
GN,  also  ZFRS  is  not  <-ZFGN;  and  so  ZFRS 
and  ZRFM+ZFRS=ZGFM+Z 


14        SCIENCE  ABSOLUTE  OF  SPACE. 

FGN=st.Z.  Therefore  also  ZDCP+ZCDQ 
=st.Z. 

II.  If  R  falls  without  the  sect  FG;  then 
ZNGR=ZMFR,  and  let  MFGN^NGHL= 
LHKO,  and  so  on,  until  FK>FR  or  begins  to 
be  >FR.  Then  KO  II  HL  II FM  (§7). 

If  K  falls  on  R,  then  KO  falls  on  RS  (§1); 
and  so  ZRFM+ZFRS=ZKFM+ZFKO=Z 
KFM+ZFGN=st.Z;  but  if  R  falls  within  the 
sect  HK,  then  (by  I)  ZRHL+^KRS=st.Z  = 
ZRFM+ZFRS=ZDCP+ZCDQ. 

§  14.  If  BN  II  AM,  and  CP  II  DQ,  and  ZBAM 
+ZABN<st.Z;  then  also  ZDCP+ZCDQ< 
st.Z. 

For  if  ZDCP+ZCDQ  were  not  <st.Z,  and 
so  (by  §1)  were  =st.^,  then  (by  §13;  also  Z. 
BAM+ZABN=st.Z  (contra  hyp.). 

§  15.  Weighing  §§  13  and  14,  the  System  of 
Geometry  resting  on  the  hypothesis  of  the 
truth  of  Euclid's  Axiom  XI  is  called  j/  and 
the  system  founded  on  the  contrary  hypoth- 
esis  is  S. 

All  things  which  are  not  expressly  said  to 
be  in  ^  or  in  S,  it  is  understood  are  enunci- 
ated absolutely ,  that  is  are  asserted  true 
whether  I  or  S  is  reality. 


SCIENCE  ABSOLUTE  OP  SPACE.        15 

§16.    If  AM  is  the  axis  of  any  L;  then  L, 
in  I  is  a  straight  J_  AM. 

N  M  F  For  suppose  BN  an  axis  from  any 
point  B  of  L;  in  i,  ZBAM+ZABN 
=st.Z,  and  so  ZBAM=rt.Z. 

And   if    C    is    any    point    of   the 
straight    AB,    and    CPU  AM;    then 
Jc(by  §  13)  CP^AM,  and  so  C  on  L 


B  A 

FIG.  12 

But  in  S,  no  three  points  A,  B,  C  on  L  or 
on  F  are  in  a  straight.  For  some  one  of  the 
axes  AM,  BN,  CP  (e.  g.  AM)  falls  between 
the  two  others;  and  then  (by  §  14)  ZBAM  and 
ZCAM  are  each  <rt.Z. 

§  17.  L  in  S  also  is  a  Line,  and  F  a  sur- 
face. For  (by  §  11)  any  plane  _*_  to  the  axis 
ray  AM  (through  any  point  of  F)  cuts  F  in 
[the  circumference  of]  a  circle,  of  which  the 
plane  (by  §  14)  is  J_  to  no  other  axis  ray  BN. 
If  we  revolve  F  about  BN,  any  point  of  F  (by 
§  12)  will  remain  on  F,  and  the  section  of  F 
with  a  plane  not  J_  ray  BN  will  describe  a  sur- 
face; and  whatever  be  the  points  A,  B  taken 
on  it,  F  can  so  be  congruent  to  itself  that  A 
falls  upon  B  (by  §  12) ;  therefore  F  is  a  uni- 
form surface. 


16 


SCIENCE  ABSOLUTE  OP  SPACE. 


Hence  evidently  (by  §§  11  and  12)  L  is  a  uni- 
form line.* 

§  18.  The  intersection  with  F  of  any  plane, 
drawn  through  a  point  A  of  F  obliquely  to  the 
axis  AM,  is,  in  S,  a  circle. 

For  take  A,  B,  C,  three  points  of  this  sec- 
tion, and  BN,  CP;  axes;  AMBN  and  AMCP 
make  an  angle,  for  otherwise  the  plane  deter- 
mined by  A,  B,  C  (from  §  16)  would  contain 
AM,  (contra  hyp.).  Therefore  the  planes  bi- 
secting J_  the  sects  AB,  AC  intersect  (§  10)  in 
some  axis  ray  FS  (of  F),  and  FB=FA=FC. 

MakeAHlFS,  and  re- 
volve FAH  about  FS;  A 
will  describe  a  circle  of 
radius  HA,  passing 
through  B  and  C,  and  sit- 
uated both  in  F  and  in 
the  plane  ABC;  nor  have 
F  and  the  plane  ABC  any- 
thing in  common  but  O  HA  (§16). 

It  is  also  evident  that  in  revolving  the  por- 
tion FA  of  the  line  L  (as  radius)  in  F  around 
F,  its  extremity  will  describe  O  HA. 


*  It  is  not  necessary  to  restrict  the  demonstration  to  the  system 
S;  since  it  may  easily  be  so  set  forth,  that  it  holds  absolutely  for 
S  and  for  I. 


SCIENCE  ABSOLUTE  OP  SPACE.        17 

j  19.  The  perpendicular  BT  to  the  axis 
BN  of  L  (falling  in  the  plane  of  L)  is,  in  S, 
tangent  to  L.  For  L  has  in  ray 
BT  no  point  except  B  (§14), 
but  if  BQ  falls  in  TBN,  then 
the  center  of  the  section  of  the 
-Q  plane  through  BQ  perpendicular 
to  TBN  with  the  F  of  ray  BN 
FIG.  14.  (§  18)  is  evidently  located  on  ray 
BQ;  and  if  sect  BQ  is  a  diameter,  evidently 
ray  BQ  cuts  in  Q  the  line  L  of  ray  BN. 

§  20.  Any  two  points  of  F  determine  a  line 
L  (§§  11  and  18);  and  since  (from  §§  16  and  19) 
L  is_L  to  all  its  axes,  every  Z  of  lines  L  in  F  is 
equal  to  the  ^  of  the  planes  drawn  through  its 
sides  perpendicular  to  F. 

§21.  Two  L  form  lines,  ray  AP  and  ray 
BD,  in  the  same  F,  making  with 
a  third  L  form  AB,  a  sum  of  inte- 


rior angles  <st.Z,  intersect. 

(By   line    AP    in    F,   is    to   be 
understood    the    line    L    drawn 
through  A  and  P,  but  by  ray  AP 
that  half  of  this  line  beginning  at  A,  in  which 
P  falls.} 

For  if  AM,  BN  are  axes  of  F,  then  the  hemi- 
planes  AMP,  BND  intersect  (§  9) ;  and  F  cuts 


18        SCIENCE  ABSOLUTE  OF  SPACE. 

their  intersection  (by  §§  7  and  11);  and  so  also 
ray  AP  and  ray  BD  intersect. 

From  this  it  is  evident  that  Euclid's  Axiom 
XI  and  all  things  which  are  claimed  in  geome- 
try and  plane  trigonometry  hold  good  abso- 
lutely in  F,  L  lines  being  substituted  in  place 
of  straights:  therefore  the  trigonometric 
functions  are  taken  here  in  the  same  sense  as 
in  .1;  and  the  circle  of  which  the  L  form  ra- 
dius =  r  in  F,  is  =  2*r;  and  likewise  area  of 
Or  (in  F)  =  -r*  (by  -  understanding  ^Ol  in  F, 
or  the  known  3.1415926.  .  .) 

§  22.  If  ray  AB  were  the  L  of  ray  AM,  and 
C  on  ray  AM;  and  the  ZCAB  (formed  by  the 
straight  ray  AM  and  the  L  form 
line  ray  AB),  carried  first  along 
the  ray  AB,  then  along  the  ray 
BA,  always  forward  to  infinity: 
the  path  CD  of  C  will  be  the 
line  L  of  CM. 

For  let  D  be  any  point  in  line 
CD  (called  later  L'),  let  DN  be  II  CM,  and  B 
the  point  of  L  falling  on  the  straight  DN.  We 
shall  have  BN=^  AM,  and  sect  AC=sect  BD,  and 
so  DN  ^  CM,  consequently  D  in  I/ .  But  if  D  in 
I/  and  DN  II  CM,  and  B  the  point  of  L  on  the 
straight  DN;  we  shall  have  AM^BN  and  CM 
whence  manifestly  sect  BD=sect  AC, 


SCIENCE  ABSOLUTE  OP  SPACE.        19 

and  D  will  fall  on  the  path  of  the  point  C,  and 
L'  and  the  line  CD  are  the  same.  Such  an  L'  is 
designated  by  I/lliL. 

§  23.  If  the  L  form  line  CDF  |||  ABE  (§  22), 
and  AB-BE,  and  the  rays  AM,  BN,  EP  are 
axes;  manifestly  CD=DF;  and  if  any  three 
points  A,  B,  E  are  of  line  AB,  and  AB=n.CD, 
we  shall  also  have  AE^n.CF;  and  so  (mani- 
festly even  for  AB,  AE,  DC  incommensurable), 
AB:CD=AE'CF,  and  AB'CD  is  independent 
of  AB,  and  completely  determined  by  AC. 

This  ratio  AB:CD  is  designated  by  the  cap- 
ital letter  (as  X)  corresponding  to  the  small  let- 
ter (as  x)  by  which  we  represent  the  sect  AC. 

_y 

§  24.    Whatever  be  x  and>v   (§23),  Y=XX. 

For,  one  of  the  quantities  x,  y  is  a  multiple 
of  the  the  other  (e.  g.  y  of  x),  or  it  is  not. 

If^=n.^,  take*=AC  =  CG=GH=&c.,  until 
we  get  AH=jF. 

Moreover,  take  CD  |||  GK  |||  HL. 

We  have  ((§23)  X=AB:CD  =  CD:GK=GK: 
HL;  and  so  AB=  f  AB 

y       HL~  [  CD 
or  Y=Xn=Xx. 

If  x}  y  are  multiples  of  /,   suppose 
and  y^ni;  (by  the  preceding)   X=Im,   Y=P, 
consequently  n       _y 

Y-Xm=Xx 


20        SCIENCE  ABSOLUTE  OF  SPACE. 

The  same  is  easily  extended  to  the  case  of 
the  incommensurability  of  x  and  y. 
But  if  q=y-x,  manifestly  Q=Y'X. 
It  is  also  manifest  that  in  J,  for  any  x,  we 
have  X=l,  but  in  S  is  X>1,  and  for  any  AB  [ 
and  ABE  there  is  such  a  CDF  |||  AB,  that  CDF 
-AB,    whence    AMBN^AMEP,    though    the 
first  be  any  multiple  of  the  second;   which  in- 
deed is  singular,  but  evidently  does  not  prove 
the  absurdity  of  S. 

§  25.  In  any  rectilineal  triangle,  the  cir- 
cles with  radii  equal  to  its  sides  are  as  the 
sines  of  the  opposite  angles. 

^  For  take  ZABC=rt.Z, 
and  AMlBAC,  and  BN  and 
CP  II  AM;  we  shall  have  CAB 
1  AMBN,  and  so  (since  CB_|_ 
BA),  CBlAMBN,  conse- 
quently CPBNl  AMBN. 

Suppose  the  F  of  ray  CP 
FIG.  17.  cuts  the   straights    BN,    AM 

respectively  in  D  and  E,  and  the  bands  CPBN, 
CPAM,  BNAM  along  the  L  form  lines  CD, 
CE,  DE.  Then  (§20)  ZCDE^the  angle  of 
NDC,  NDE,  and  so  =  rt.Z;  and  by  like  reason- 
ing Z  CED  =  Z  CAB.  But  (by  §  21)  in  the  L  line 
A  CDE  (supposing  always  here  the  radius  =1), 
EC:D€=l:sin  DEC =1:  sin  CAB. 


SCIENCE  ABSOLUTE  OF  SPACE.        21 

Also  (by  §  21) 

EC:DC=OEC:ODC(inF)=OAC:OBC  (§18); 
and  so  is  also 

0AC:OBC  =  l:sin  CAB; 

whence  the  theorem  is  evident  for  any  triangle. 
§  26.    In  any  spherical  triangle,  the  sines 
of  the  sides  are  as  the  sines  of  the  angles 
opposite. 

For  take  ZABC=rt.Z,  and 
CED 1  to  the  radius  OA  of  the 
sphere.  We  shall  have  CED  \_ 
AOB,  and  (since  also  BOC  \_ 
BOA),  CDlOB.  But  in  the 
no.  is.  triangles  CEO,  CDO  (by  §  25) 
OEC:OOC:ODC=sin  COE  :  1  •  sin  COD=sin 
AC  :  1  :  sin  BC;  meanwhile  also  (§  25)  OEC  : 
ODC=sin  CDE  :  sin  CED.  Therefore,  sin 
AC  :  sin  BC=sin  CDE  :  sin  CED;  but  CDE= 
rt.Z  =  CBA,  and  CED  =  CAB.  Consequently 

sin  AC  :  sin  BC=1  :  sin  A. 
Spherical  trigonometry ,  flowing  from  this, 
is  thus  established  independently  of  Axiom 
AY. 

§  27.    If  AC  and  BD  are  J_  AB,  and  CAB  is 
carried  along  the  straight  AB;  we  shall  have, 
designating  by  CD  the  path  of  the  point  C, 
CD  :  AB  =  sin  u  :  sin  v. 


22        SCIENCE  ABSOLUTE  OF  SPACE. 


For  take  DElCA; 
in  the  triangles  ADE, 
ADB  (by  §  25) 
OED : OAD : OAB= 

sin  u  :  1  :  sin  v. 


G  F 

FIG.  19.  In    revolving    BACD 

about  AC,  B  describes  OAB,  and  D  describes 
OED;  and  designate  here  by  sOCD  the  path 
of  the  said  CD.     Moreover,  let  there  be  any  [123 
polygon  BFG. . .  inscribed  in  OAB. 

Passing  through  all  the  sides  BF,  FG,  &c., 
planes  J_  to  OAB  we  form  also  a  polygonal  fig- 
ure of  the  same  number  of  sides  in  sOCD,  and 
we  may  demonstrate,  as  in  §  23,  that  CD  :  AB 
=DH  :  BF=HK  :  FG,  &c.,  and  so 

DH+HK  &c.  :  BF+FG  &c.  :  =CD  :  AB. 

If  each  of  the  sides  BF,  FG . .  .  approaches 
the  limit  zero,  manifestly 

BF+FG+ .  .  .  =0  AB         and 
DH+HK+...=OED. 

Therefore  also  OED  :  OAB=CD  :  AB.  But 
we  had  OED  :  OAB=sin  u  :  sin  v.  Conse- 
quently 

CD  :  AB=sin  u  :  sin  v. 

If  AC  goes  away  from  BD  to  infinity,  CD  : 
AB,  and  so  also  sin  u  :  sin  v  remains  constant; 
but  ^=rt.  Z  (§1),  and  if  DM  II BN,  v=z; 
whence  CD  :  AB=1  :  sin  z. 


SCIENCE  ABSOLUTE  OF  SPACE.        23 


The  path  called  CD  will  be  denoted  by  CD 
illAB. 

§  28.    If  BN  II  ^  AM,  and  C  in  ray  AM,  and 
AC=x:  we  shall  have  (§  23) 

X=sin  u  :  sin  v. 
For  if  CD  and  AE  are  _L  BN, 
*<\F       /    f^  and  BF_L  AM;  we  shall  have  (as 
in  §  27) 

OBF  :  ODC^sin  u  :  sin  v. 
ButevidentlyBF=AE:  therefore 

OEA  :  OCD=sin  u  :  sin  v. 
But  in  the  F  form  surfaces  of 
AM  and  CM  (cutting  AMBN  in  AB  and  CG) 
(by  §21) 

OEA  :  ODC=AB  :  CG=X. 
Therefore  also 


M 

FIG.  20. 


29. 


X=sin  u 
If  ZBAM=rt. 


sin  v. 

and  sect  AB—jy,  and 
BNIIAM,  we  shall 
have  in  S 

Y—cotan  ^  u. 
For,  if  sect  AB- 
sect  AC,   and  CPU 
AM  (and  so  BNll  ^ 
CP),    and    ZPCD  = 
ZQCD;  there  is  given  (§19)   DS_j_ray  CD,  so 
that  DS  II  CP,  and  so  (§  1)  DT  II  CQ.    Moreover, 
if  BE  1  ray  DS,  then  (§  7)  DS  II  BN,  and  so  (§  6) 


FiG.  21. 


24 


SCIENCE  ABSOLUTE  OF  SPACE. 


BNnES,  and  (since  DT  II  CG)  BQllET;  con- 
sequently (§1)  ZEBN=ZEBQ.  Let  BCF  be 
an  1,-line  of  BN,  and  FG,  DH,  CK,  EL,  L  form 
lines  of  FT,  DT,  CQ  and  ET;  evidently  (§22) 
HG=DF=DK-HC;  therefore, 
CG=2CH=2z>. 

Likewise  it  is  evident  BG=2BL=2^. 
ButBC=BG-CG;    wherefore  y=z~v,  and 
so  (§24)  Y=Z:V. 
Finally  (§  28) 

Z  =  l  :  sin  y2  u, 
and  V=l  :  sin  (rt.^~X  ^)> 
consequently    Y=cotan  ^  u. 

§  30.  However,  it  is  easy  to  see  (by  §  25) 
that  the  solution  of  the  problem  of  Plane 
Trigonometry,  in  S,  requires 
the  expression  of  the  circle 
in  terms  of  the  radius;  but 
this  can  by  obtained  by  the 
rectification  of  L. 

Let  AB,  CM,  C'M'  be  _L 
ray  AC,  and  B  anywhere  in 
ray  AB;  we  shall  have  (§  25) 

sin  u  :  sin  v=Qp  :  Qy, 
and  sin  u'  :  sin  v1  '  =Qp'  : 


FIG.  22. 


and  so 


sm  v 


sin  v 


SCIENCE  ABSOLUTE  OP  SPACE.        25 


But  (by  §  27)  sin  v  :  sin  v'  =cos  u  :  cos  u  ; 

,,    sin  u  _        sin      ' 
consequently-     —  .Qy= 


COS  U  COS  U 

or     OjF  :  Oy'  =tan  u' :  tan  u=tan  w  :  tan  w' . 

Moreover,  take  CN  and  C'N'  II  AB,  and  CD, 
C'D'  L-form  lines  j_  straight  AB;  we  shall 
have  also  (§21) 

Qy  :  Qy'  =r  :  r' ,  and  so 

r  :  r  =tan  w  :  tan  w' . 

Now  let  p  beginning  from  A  increase  to  in- 
finity; then  w=z,  and  w  '=z ' ,  whence  also 
r  :  r  —tan  z  :  tan  z ' . 

Designate  by  i  the  constant 

r  :  tan  z  (independent  of  r) ; 

whilst     y=0, 

r     i  tan  z  ,  ^         -. 

-=-         — =1,  and  so 

y       y 

-2—^i.    From  §29,  tan  z=y2  (Y-Y-1); 
tan  z 

therefore  -^_^=i^ 

or  (§24)  ^_L.i. 

~\   i~ 

But  we  know  the  limit  of  this  expression 
(where  y=0)  is 

/I 

-.     Therefore 

nat.  log  I 


26        SCIENCE  ABSOLUTE  OF  SPACE. 

-T=*,  and 
nat.   log  I 

1=0=2.7182818..., 
which  noted  quantity  shines  forth  here  also. 

If  obviously  henceforth  i  denote  that  sect  of 
which  the  1=0,  we  shall  have 

r—i  tan  z. 
But  (§  21)  Gy=2-r/  therefore 

=2«*  tan  z=«i     T-Y-'  =  * 


(by  §24). 


nat.  log  Y 

§  31.  For  the  trigonometric  solution  of  all 
right-angled  rectilineal  triangles  (whence  the 
resolution  of  all  triangles  is  easy  ,  in  S,  three  LI 
equations  suffice  :  indeed  (a,  b  denoting  the 
sides,  c  the  hypothenuse,  and  «,  ,3  the  angles 
opposite  the  sides)  an  equation  expressing  the 
relation 

1st,  between  a,  c,  a; 

2d,    between  a,  «,  £; 

3d,    between  a,  b,  c; 
of  course  from  these  equations 
emerge  three  others  by  elim- 
ination. FIG.  23. 

From  §§  25  and  30 

1  :  sin  a=(C-C~l)  :  (A-A-])  = 

f_^r  j   (equation  for  c,  a  and  «). 


SCIENCE  ABSOLUTE  OF  SPACE.        27 

II.    From  §  27  follows  (if  ,?M  II  rN) 
cos  «:  sin  ,3=1  :  sin  2/y  but  from  §  29 
1  :sin  ^ 


therefore  cos  a  sin  ^ 
(equation  for  «,  /?  and  <z). 

III.    If  0.0.  'JL/5«r>  and  A9'  and  rr'llaa/  (§  27), 
and  ,3'a>'  J_aa';  manifestly  (as  in  §27) 


sin 


,  or 

*  ~-5  if  5.  ~^ 


(equation  for  #,  <5  and  c). 
If  r«fl=rt.^,  and  /35_L«^; 

O<^  '  O<^=1  *  sin  «,  and 
Qc  :  Q(d=pd\=  1  :  cos  a, 

and  so  (denoting  by  O^2?  for  any  x,  the  product 
manifestly. 


But  (by  §  27  and  II) 

Qd=Q6.^(A+A~1),  consequently 

fL      -_?}  ~_I/    f    5L     -^1  2  f     b      -b^  ~        f     a_  •  -a^  2 

another  equation  for  #,  <5  and  c  (the  second 


28        SCIENCE  ABSOLUTE  OF  SPACE. 

member  of  which  may  be  easily  reduced  to  a 
form  symmetric  or  invariable).  [15 

Finally,  from 

— — -^(A+A"1),  and  ~ — -^(B+B"1),  we  get 
sin  p  sin  a 

(by  III) 

COt  a  COt  fi=% 

(equation  for  «,  /5,  and  c. 

§  32.  It  still  remains  to  show  briefly  the 
mode  of  resolving  problems  in  S,  which  being 
accomplished  (through  the  more  obvious  exam- 
ples), finally  will  be  candidly  said  what  this 
theory  shows. 

I.  Take  AB  a  line  in  a  plane,  and  y=f(x] 
its  equation  in  rectangular  co- 
ordinates, call  dz  any  increment 
of  z,  and  respectively  dx,  dy,  du 
the  increments  of  x,  of  y,  and  of 
)A  the  area  u,  corresponding  to 

FIG.  24. 

BH 


this  dz;  take   BH  III  CF,  and  ex- 


press  (from  §  31)  -^—  by  means  of  y ,  and  seek 
ax* 

the  limit  of  -¥•    when  dx  tends    towards   the 
dx 

limit  zero  (which  is  understood  where  a  limit 
of  this  sort  is  sought) :  then  will  become  known 

also  the  limit  of  — ^.,   and  so  tan  HBG;   and 


SCIENCE  ABSOLUTE  OF  SPACE.        29 

(since  HBC  manifestly  is  neither  >  nor  <,  and 
so  =rt.  Z),  the  tangent  at  B  of  BG  will  be  de- 
termined by  y. 

II.     It  can  be  demonstrated 

~ 


Hence  is  found  the  limit  of  -,  and  thence, 

ax 

by  integration,  z  (expressed  in  terms  of  x. 

And  of  any  line  given  in  the  ^concrete,  the 
equation  in  S  can  be  found;  e.  g.,  of  L.  For 
if  ray  AM  be  the  axis  of  L;  then  any  ray  CB 
from  ray  AM  cuts  L  [since  (by  §  19)  any 
straight  from  A  except  the  straight  AM  will 
cut  L]  ;  but  (if  BN  is  axis) 

X=l:sin  CBN  (§28), 
and  Y-cotan  y>  CBN  (§29),  whence 


or  y 


the  equation  sought. 
Hence  we  get 


ax 

and         =1  :  sin  CBN-X;  and  so 
ax 

dy  —  (X3-!)^- 
BH~l 


30        SCIENCE  ABSOLUTE  OP  SPACE. 


^=X(X2-l),  and 


=X2(X2-1),   whence,   by  inte- 
ctx 

gration,  we  get  (as  in  §  30) 

^=^(X8-l)*=*cot  CBN. 
III.     Manfestly 


dx          dx 

which  (unless  given  in  y)  now  first  is  to  be  ex- 
pressed in  terms  of  y;  whence  we  get  u  by 
integrating. 

D  If  AB=/,  AC=?,  CD=r,  and 
CABDC=s/  we  might  show  (as 
in  II)  that 

-= —=r,  which  = 

FIG.  25.        aq 

and,  integrating,  s=%pi    ^f_ ~f 

This  can  also  be  deduced  apart  from  inte- 
gration. 

For  example,  the  equation  of  the  circle  (from 
§  31,  III),  of  the  straight  (from  §  31,  II),  of  a 
conic  (by  what  precedes),  being  expressed,  the 


SCIENCE  ABSOLUTE  OF  SPACE.       31 

areas  bounded  by  these  lines  could  also  be  ex- 
pressed. 

We  know,  that  a  surface  t,  \\\  to  a  plane  fig- 
ure/* (at  the  distance  q],  is  to/>  in  the  ratio  of 
the  second  powers  of  homologous  lines,  or  as 
f   4     -±\  2 

I^    |^i_£?i  J       :  1. 

It  is  easy  to  see,  moreover,  that  the  calcula- 
tion of  volume,  treated  in  the  same  manner, 
requires  two  integrations  (since  the  differen- 
tial itself  here  is  determined  only  by  integra- 
tion) ;  and  before  all  must  be  investigated  the  [in 
volume  contained  between  p  and  /,  and  the  ag- 
gregate of  all  the  straights  A-p  and  joining 
the  boundaries  of  p  and  t. 

We  find  for  the  volume  of  this  solid  (whether 
by  integration  or  without  it) 

f     2q         ^ 


The  surfaces  of  bodies  may  also  be  deter- 
mined in  S,  as  well  as  the  curvatures,  the 
involutes,  and  evolutes  of  any  lines,  etc. 

As  to  curvature;  this  in  S  either  is  the  curv- 
ature of  L,  or  is  determined  either  by  the 
radius  of  a  circle,  or  by  the  distance  to  a 
straight  from  the  curve  |||  to  this  straight;  since 
from  what  precedes,  it  may  easily  be  shown, 
that  in  a  plane  there  are  no  uniform  lines  other 
than  L-lines,  circles  and  curves  |||  to  a  straight. 


32        SCIENCE  ABSOLUTE  OF  SPACE. 

IV.     For  the  circle  (as  in  III)   -  area  G 
,  whence  (by  §29),  integrating, 


dx 


V.    For  the  area  CABDC=^  (inclosed  by  an 
M        N     L,  form  line  AB=r,  the  III  to  this, 
CV=j>,  and  the  sects  AC=BD=^) 

— —=y;  and   (§  24)    y— re^,   and  so 

D    «* 

(integrating)  ^=r/  M_^T 

If  x  increases  to  infinity,  then,  in 


FIG.  26.  -2* 

S,  ^i=^0,  and  so  u=ri.    By  the  size 

of  MABN,  in  future  this  limit  is  understood. 
In  like  manner  is  found,  if  p  is  a  figure  on 
F,  the  space  included  by^>  and  the  aggregate 
of  axes  drawn  from  the  boundaries  of  p  is 
equal  to  Y*pi. 

VI.  If  the  angle  at  the  cen- 
ter of  a  segment  z  of  a  sphere 
is  2u,  and  a  great  circle  is^>, 
and  x  the  arc  FC  (of  the  angle 

«);  (§25) 

l:sin  ^=/:QBC, 
and  hence  OBC=/>  sin  u. 

Meanwhile  *=^,  and  dx^^- 

2^  2* 


FIG.  27. 


SCIENCE  ABSOLUTE  OF  SPACE.       33 


Moreover,  -=OBC,  and  hence 

ax    . 

j  /2 

-j-==±£-  sin  u,  whence  (integrating) 

au     2- 

_ver  sin  u  ,2 
~2*        • 

The  F  may  be  conceived  on  which  P  falls 
(passing  through  the  middle  F  of  the  seg- 
ment) ;  through  AF  and  AC  the  planes  FEM, 
CEM  are  placed,  perpendicular  to  F  and  cut- 
ting F  along  FEG  and  CE;  and  consider  the 
L  form  CD  (f  rom  C  1  to  FEG),  and  the  L  form 
CF;  (§20)  CEF=^,  and  (§21) 

gg=ver  sin  *   and  so  ^=FD./. 

P  47T 

But  (§21)/=^.FGD;  therefore 

^=r:.FD.FDG.     But  (§21) 
FD.FDG=FC.FC;  consequently 
^=7r.FC.FC=areaoFC,  in  F. 

Now  let  BJ=CJ=r/  (§30) 
2r=*(Y—  Y-1),   and   so   (§21) 
area  O2r  (in  F)  =^2(Y-  Y-1;2. 
Also  (IV) 


area 

therefore,  area  O2^  (in  F)  =area  Q2y1  and  so 
the  surface  z  of  a  segment  of  a  sphere  is 
equal  to  the  surface  of  the  circle  described 
with  the  chord  FC  as  a  radius. 


34        SCIENCE  ABSOLUTE  OF  SPACE. 
Hence  the  whole  surface  of  the  sphere 


and  the  surfaces  of  spheres  are  to  each  other 
as  the  second  powers  of  their  great  circles. 

VII.    In  like  manner,   in  S,   the  volume  of 
the  sphere  of  radius  x  is  found 


the  surface  generated  by  the  rev- 
olution of  the  line  CD  about  AB 


A    T~     B  and  the  body  described  by  CABDC 
FIG.  29.  =  i/_v2^/r) Q~M2 

But  in  what  manner  all  things  treated 
from  (IV]  even  to  here,  also  may  be  reached 
apart  from  integration,  for  the  sake  of  brev- 
ity is  suppressed. 

It  can  be  demonstrated  that  the  limit  of 
every  expression  containing  the  letter  i  (and 
so  resting  upon  the  hypothesis  that  i  is  given),  [19] 
when  i  increases  to  infinity,  expresses  the 
quantity  simply  for  I  (and  so  for  the  hypoth- 
esis of  no  i),  if  indeed  the  equations  do  not  be- 
come identical. 

But  beware  lest  you  understand  to  be  sup- 
posed, that  the  system  itself  may  be  varied 
(for  it  is  entirely  determined  in  itself  and  by 
itself) ;  but  only  the  hypothesis,  which  may  be 


SCIENCE  ABSOLUTE  OF  SPACE.        35 

done  successively,  as  long  as  we  are  not  con- 
ducted to  an  absurdity.  Supposing  therefore 
that,  in  such  an  expression,  the  letter  i,  in 
case  S  is  reality,  designates  that  unique  quan- 
tity whose  \~e;  but  if  r  is  actual,  the  said 
limit  is  supposed  to  be  taken  in  place  of  the 
expression  :  manifestly  all  the  expressions  or- 
iginating from  the  hypothesis  of  the  reality 
ofS  (in  this  sense]  will  be  true  absolutely, 
although  it  be  completely  unknown  whether 
or  not  I  is  reality 

So  e.  g.  from  the  expression  obtained  in  §  30 
easily  (and  as  well  by  aid  of  differentiation  as 
apart  from  it)  emerges  the  known  value  in  J, 


from  I  (§  31)  suitably  treated,  follows 

1  :  sin  a—C  :  a; 
but  from  II 

COS 


sin  - 


^ 

=  1,  and  so 


the  first  equation  in  III  becomes  identical,  and 
so  is  true  in  I,  although  it  there  determines 
nothing;  but  from  the  second  follows 


These  are  the  known  fundamental  equa- 
tions of  plane  trigonometry  in  I. 


36        SCIENCE  ABSOLUTE  OF  SPACE. 

Moreover,  we  find  (from  §  32)  in  r,  the  area 
and  the  volume  in  III  each  =pq;  from  IV 

area  O#=-^; 
(from  VII)  the  globe  of  radius  x 

=%xxs,  etc. 

The  theorems  enunciated  at  the  end  of  VI 
are  manifestly  true  unconditionally. 

§  33.  It  still  remains  to  set  forth  (as  prom- 
ised in  §  32)  what  this  theory  means. 

I.  Whether  I  or  some  one  S  is  reality,  re- 
mains undecided. 

II.  All  things  deduced  from  the  hypothesis 
of  the  falsity  of  Axiom  XI  (always  to  be  un- 
derstood in  the  sense  of  §  32)   are  absolutely 
true,  and  so  in  this  sense,   depend  upon  no 
hypothesis. 

There  is  therefore  a  plane  trigonometry  a 
priori,  in  which  the  system  alone  really  re- 
mains  unknown;  and  so  where  remain  un- 
known solely  the  absolute  magnitudes  in  the 
expressions,  but  where  a  single  known  case 
would  manifestly  fix  the  whole  system.  But 
spherical  trigonometry  is  established  abso- 
lutely in  §  26. 

(And  we  have,  on  F,  a  geometry^wholly  an-" 
alogous  to  the  plane  geometry  of  J.) 

III.  If  it  were  agreed  that  -  exists,  nctthing 
more  would  be  unknown  in  this  respect;  but 


SCIENCE  ABSOLUTE  OF  SPACE.        37 

if  it  were  established  that  I  does  not  exist, 
then  (§  31),  (e.  g.)  from  the  sides  x,  y>  and  the 
rectilineal  angle  they  include  being  given  in  a 
special  case,  manifestly  it  would  be  impossible 
in  itself  and  by  itself  to  solve  absolutely  the 
triangle,  that  is,  to  determine  a  priori  the 
other  angles  and  the  ratio  of  the  third  side  to 
the  two  given;  unless  X,  Y  were  determined, 
for  which  it  would  be  necessary  to  have  in 
concrete  form  a  certain  sect  a  whose  A  was 
known;  and  then  i  would  be  the  natural  unit 
for  length  (just  as  e  is  the  base  of  natural 
logarithms). 

If  the  existence  of  this  i  is  determined,  it 
will  be  evident  how  it  could  be  constructed, 
at  least  very  exactly,  for  practical  use. 

IV.  In  the  sense  explained  (I  and  II),  it  is 
evident  that  all  things  in  space  can  be  solved 
by  the  modern   analytic  method  (within  just 
limits  strongly  to  be  prdised). 

V.  Finally,  to  friendly  readers  will  not  be 
unacceptable;  that  for  that  case  wherein  not  I 
but  S  is   reality,   a   rectilineal  figure  is  con- 
structed equivalent  to  a  circle. 

§  34.  Through  D  we  may  draw  DM  II  AN  in 
the  following  manner.  From  D  drop  DB  J_  AN ; 
from  any  point  A  of  the  straight  AB  erect  AC 
IAN  (in  DBA),  and  let  fall  DC1AC.  We 


38        SCIENCE  ABSOLUTE  OF  SPACE. 

will  have   (§  27)    OCD  :  OAB  =  1  :  sin  z,    pro- 
M  vided  that  DM  II  BN.    But  sin 
z  is  not  >1;    and  so  AB  is 
not  >DC.   Therefore  a  quad- 
-—  rant  described  from  the  cen- 


A  B  S 

FIG.  so.  ter  A  in  BAG,  with  a  radius 

—  DC,  will  have  a  point  B  or  O  in  common  with 
ray  BD.  In  the  first  case,  manifestly  ^=rt.^; 
but  in  the  second  case  (§  25) 

(OAO-OCD)  :  OAB=1  :  sin  AOB, 
and  so  ^=AOB. 

If  therefore  we  take  ^=AOB,  then  DM  will 
be  II  BN. 

§  35.    If  S  were  reality;  we  may,  as  follows, 
draw  a  straight  _L  to  one  arm  of  an  acute  angle,  [211 
which  is  II  to  the  other. 

Take  AMI BC,  and 
suppose  AB=BC  so 
small  (by  §  19),  that 
if  We  draw  BN  II  AM 

(§    34),     ABN   >   the 
FIG.  31.  .  . 

given  angle. 

Moreover  draw  CP II  AM  (§34);  and  take 
NBG  and  PCD  each  equal  to  the  given  angle; 
rays  BG  and  CD  will  cut;  for  if  ray  BG  (fall- 
ing by  construction  within  NBC)  cuts  ray  CP 
in  E;  we  shall  have  (since  BN^CP),  ZEBC< 
ZECB,  and  so  EC<EB.  Take  EF=EC,  EFR 


SCIENCE  ABSOLUTE  OP  SPACE.        39 

=ECD,  and  FS  II  EP;  then  FS  will  fall  within 
BFR.  For  since  BN  II  CP,  and  so  BN  II  EP, 
and  BN  II  FS;  we  shall  have  (§  14) 

ZFBN+ZBFS  <  (  st.  Z  =FBN+BFR)  ; 
therefore,  BFS  <  BFR.  Consequently,  ray  FR 
cuts  ray  EP,  and  so  ray  CD  also  cuts  ray  EG 
in  some  point  D.  Take  now  DG=DC  and 
DGT=DCP=GBN;  we  shall  have  (since  CD^ 
GD)  BN^GT^CP.  Let  K  (§  19)  be  the  point 
of  the  L-f  orm  line  of  BN  falling  in  the  ray  BG, 
and  KL  the  axis;  we  shall  have  BN^KL, 
and  so  BKL=BGT=DCP;  but  also  KL*=CP: 
therefore  manifestly  K  fall  on  G,  and  GT  II  BN. 
But  if  HO  bisects  J_BG,  we  shall  have  con- 
structed HO  II  BN. 

36.  Having-  given  the  ray  CP  and  the 
plane  MAB,  take  CBlthe 
plane  MAB,  BN  (in  plane 
BCPj  J-BC,  and  CQ  II  BN 
(§  34)  ;  the  intersection  of  ray 
CP  (if  this  ray  falls  within 
B  BCQ)  with  ray  BN  (in  the 


FIG.  32.  pian    CBN),  and  so  with  the 

plane  MAB  is  found.  And  if  we  are  given 
the  two  planes  PCQ,  MAB,  and  we  have  CB 
Ito  plane  MAB,  CR  i.  plane  PCQ;  and  (in 
plane  BCR)  BN1BC,  CS_uCR,  BN  will  fall 
in  plane  MAB,  and  CS  in  plane  PCQ;  and  the 


40        SCIENCE  ABSOLUTE  OP  SPACE}. 

intersection  of  the  straight  BN  with  the 
straight  CS  (if  there  is  one)  having  been  found, 
the  perpendicular  drawn  through  this  inter- 
section, in  PCQ,  to  the  straight  CS  will  mani- 
festly be  the  intersection  of  plane  MAB  and 
plane  PCQ. 

§  37.  On  the  straight  AM  II  BN,  is  found  such 
an  A,  that  AM  ^=BN.  If  (by 
§  34)  we  construct  outside 
of  the  plane  NBM,  GT  II 
ICBN,  and  make  BG1GT, 
GC=GB,  and  CPllGT; 
and  so  place  the  hemi- 
plane  TGD  that  it  makes 
with  hemi-plane  TGB  an  angle  equal  to  that 
which  hemi-plane  PCA  makes  with  hemi-plane 
PCB;  and  is  sought  (by  §36)  the  intersection 
straight  DQ  of  hemi-plane  TGD  with  hemi- 
plane  NBD;  and  BA  is  made  -LDQ. 

We  shall  have  indeed,  on  account  of  the  sim- 
ilitude of  the  triangles  of  Iy  lines  produced  on 
the  F  of  BN  (§  21^,  manifestly  DB-DA,  and 
AM^BN. 

Hence  easily  appears  (L-lines  being  given  by 
their  extremities  alone)  we  may  also  find  a 
fourth  proportional,  or  a  mean  proportional, 
and  execute  in  this  way  in  F,  apart  from  Ax- 
iom XI,  all  the  geometric  constructions  made 


SCIENCE  ABSOLUTE  OP  SPACE. 


41 


on  the  plane  in  I.  Thus  e.  g.  a  perigon  can 
be  geometrically  divided  into  any  special  num- 
ber of  equal  parts,  if  it  is  permitted  to  make 
this  special  partition  in  Z. 

§  38.  If  we  construct  (by  §  37]  for  example, 
NBQ-^  rt.Z,  and  make  (by 
§  35),  in  S,  AMI  ray  BQ  and  I 
BN,  and  determine  (by  §37) 
IM^BN;  we  shall  have,  if  I A 
FIG.  34.  =*,(§  28),  X=l  :  sin  #  rt.  Z=2, 

and  x  will  be  constructed  geometrically. 

And  NBQ  may  be  so  computed,  that  IA  dif- 
fers from  i  less  than  by  anything  given,  which 
happens  for  sin  NBQ=V0. 

§  39.  If  (in  a  plane)  PQ  and  ST  are  I!  to  the 
straight  MN  (§27),  and  AB,  CD  are  equal 
perpendiculars  to  MN;  manifestly  ADEC= 
c  A  BE  A;  and  so  the  angles 
Q  (perhaps  mixtilinear)  ECP, 
EAT  will  fit,  and  EOEA. 
If,  moreover,  CF= AG,  then 
AACF^ACAG,  and  each 
is  half  of  the  quadrilateral 
FAGC. 

If  FAGC,  HAGK  are  two  quadrilaterals  of 
this  sort  on  AG,  between  PQ  and  ST;  their 
equivalence  (as  in  Euclid)  is  evident,  as  also 


42        SCIENCE  ABSOLUTE  OF  SPACE. 

the  equivalence  of  the  triangles  AGC,  AGH, 
standing  on  the  same  AG,  and  having  their 
vertices  on  the  line  PQ.  Moreover,  ACF  = 
CAG,  GCQ=CGA,  and  ACF+ACG+GCQ= 
st.Z  (§32);  and  so  also  CAG+ACG+CGA=  [23] 
st.  Z ;  therefore,  in  any  triangle  ACG  of  this 
sort,  the  sum  of  the  three  angles  =st.  ^.  But 
whether  the  straight  AG  may  have  fallen  upon 
AG  (which  III  MN),  or  not;  the  equivalence  of 
the  rectilineal  triangles  AGC,  AGH,  as  well 
of  themselves,  as  of  the  sums  of  their  angles, 
is  evident. 

§  40.    Equivalent    triangles   ABC,    ABD, 
^(henceforth  rectilineal),  hav- 
ing one  side  equal,  have  the 
sums  of  their  angles  equal. 
For  let  MN  bisect  AC  and 
B  BC,    and   take    (through  C) 

FIG.  36.  PQlllMN;    the  point  D  will 

fall  on  line  PQ. 

For,  if  ray  BD  cuts  the  straight  MN  in  the 
point  E,  and  so  (§  39)  the  line  PQ  at  the  dis- 
tance E)F=E}B;  we  shall  have  AABC=AABF, 
and  so  also  AABD=AABF,  whence  D  falls 
at  F. 

But  if  ray  BD  has  not  cut  the  straight  MN, 
let  C  be  the  point,  where  the  perpendicular  bi- 
secting the  straight  AB  cuts  the  line  PQ,  and 


SCIENCE  ABSOLUTE  OP  SPACE.        43 

let  GS=HT,  so,  that  the  line  ST  meets  the 
ray  BD  prolonged  in  a  certain  K  (which  it  is 
evident  can  be  made  in  a  way  like  as  in  §  4) ; 
moreover  take  SR=SA,  ROlllST,  and  O  the 
intersection  of  ray  BK  with  RO;  then  AABR 
=  AABO  (§39),  and  so  AABC>AABD  (con- 
tra hyp.). 

§  41.  Equivalent  triangles  ABC,  DEF 
have  the  sums  of  their  triangles  equal. 

,L  H     F          -^or  ^et  MN  bisect 

M — ^rV-s  ^TTT"0  AC  and  BC>  and  PQ 

4-/1Q  bisect  DF  and  FE; 
\l  \  and  take  RS  III  MN, 
~s-  E  and  TO  III  PQ ;  the  per- 
pendicular AG  to  RS 
will  equal  the  perpendicular  DH  to  TO,  or  one 
for  example  DH  will  be  the  greater. 

In  each  case,  the  ODF,  from  center  A,  has 
with  line-ray  GS  some  point  K  in  common, 
and  (§39)  AABK=AABC=ADEF.  But  the 
A  AKB  (by  §  40)  has  the  same  angle-sum  as 
ADFE,  and  (by  §  39)  as  AABC.  Therefore 
also  the  triangles  ABC,  DEF  have  each  the 
same  angle-sum. 

In  S  the  inverse  of  this  theorem  is  true. 
For  take  ABC,  DEF  two  triangles  having 
equal  angle-sums,   and  ABAL=ADEF;  these 
will  have  (by  what  precedes)  equal  angle-sums, 


44        SCIENCE  ABSOLUTE  OF  SPACE. 

and  so  also  will  AABC  and  AABL,  and  hence 
manifestly 

BCL+BLC+CBL=st.  Z. 

However  (by  §  31),  the  angle-sum  of  any  tri-  \M\ 
angle,  in  S,  is  <st.Z. 
Therefore  L,  falls  on  C. 

§  42.    Let  u  be  the  supplement  of  the  angle- 
sum  of  the  AABC,  but  v  of  ADEF;   then  is 
:z;. 

For  if  p  be  the  area  of  each 
of  the  triangles  ACG,  GCH, 
HCB,    DFK,    KFE;    and 
AABC=m./,  and  ADEF= 
A~G    H~*B  n.p;  and  5  the  angle-sum  of 
FIG.  38.  any  triangle  equivalent  top; 

manifestly 

st.  ^-—u—m.s  —  (m  —  l)st.  Z  =st.  Z  —  /^(st.  Z.  —s}  ; 
and   u=m(st.^—  5);    and    in    like   manner  V— 


Therefore  AABC  :  ADEF=m  :  n=u\v. 

It  is  evidently  also  easily  extended  to  the 
case  of  the  incommensurability  of  the  triangles 
ABC,  DEF. 

In  the  same  way  is  demonstrated  that  tri- 
angles on  a  sphere  are  as  the  excesses  of  the 
sums  of  their  angles  above  a  st.<. 

If  two  angles  of  the  spherical  A  are  right, 
the  third  z  will  be  the  said  excess.  But 


SCIENCE  ABSOLUTE  OF  SPACE.        45 


(a  great  circle  being  called  p)  this  A  is  mani- 
festly 

=|-  1  (§32,  VI);       ' 

consequently,  any  triangle  of  whose  angles  the 
excess  is  z,  is 


A 


§  43.    Now,  in  S,  the  area  of  a  rectilineal  A 
is  expressed  by  means  of  the  sum  of  its  angles. 
M  N'  If  AB  increases  to  infinity; 

(§  42)     A  ABC  :  (rt.^-u-v) 
7  will  be  constant.    But  A  ABC 
i  =BACN  (§  32,  V),  and  rt.Z 
;  —u—v=z     (§    1);      and     so 
:   BACN  :  ^=AABC  :  (rt.  Z- 


D  '     Moreover,  manifestly  (§  30) 
FIG. 39.  BDCN  :  BD'C'N'-/:  r' ~- 

tan  z  :  tan  z' . 

But  for  y'=o,  we  have 

BD'C'N'  ,  tan*'  , 

BAC'N' =  '  a  °  — ~^~ 

consequently, 

BDCN  :  BACN=tan  z  :  z. 
But  (§32) 

BDCN=r.*'=*'2  tan  ^y 
therefore, 


46 


SCIENCE  ABSOLUTE  OF  SPACE. 


M 


A 

FIG.  40. 


Designating  henceforth,  for  brevity,  any  tri- 
angle the  supplement  of  whose  angle-sum  is  z 
by  A,  we  will  therefore  have  A  —z.i^1. 

Hence  it  readily  flows 
that,     if     OR  II  AM    and 
RO||AB,  the  area  com- 
prehended between  the 
0  straights  OR,   ST,   BC  D»: 

(which  is  manifestly  the 
absolute  limit  of  the  area  of  rectilineal  tri- 
angles increasing  without  bound,  or  of  A  for 
^=st.^),  is  —T,^—  area  0^,  in  F. 

This  limit  being  denoted  by  n ,  moreover 
(by  §  30)  7rr2=tan2^.n  =  area  £>r  in  F  (§  21)  = 
area  ®s  (by  §32,  VI)  if  the  chord  CD  is  called  s. 
If  now,  bisecting  at  right  angles  the  given 
radius  5  of  the  circle  in  a  plane  (or  the  Iy  form 
radius  of  the  circle  in  F),  we  construct  (by 
§  34)  DB|^CN;  by  dropping  CA  1  DB,  and 
erecting  CM  1  CA,  we  shall 
1  get  z;  whence  (by  §  37),  assum- 
ing at  pleasure  an  L  form 
radius  for  unity,  tan2^  can  be 
determined  geometrically  by 
means  of  two  uniform  lines 
A_  of  the  same  curvature  (which, 
3  their  extremities  alone  being 
given  and  their  axes  con- 


FIG.  41. 


SCIENCE  ABSOLUTE  OF  SPACE.       47 

structed,  manifestly  may  be  compared  like 
straights,  and  in  this  respect  considered  equiv- 
alent to  straights) . 

Moreover,  a  quadrilateral,  ex.  gr.  regular 
=  n  is  constructed  as  follows: 

Take  ABC=rt.Z,  BAC=i  rt. 
Z,  ACB=±  rt.  Z,  and  BC=*. 

By  mere  square  roots,  X  (from 
§  31,  II)  can  be  expressed  and  (by 
§  37)  constructed;  and  having  X 
(by  §  38  or  also  §§  29  and  35),  x  itself  can  be 
determined.  And  octuple  A  ABC  is  manifestly 
=  n ,  and  by  this  a  plane  circle  of  radius  s  is 
geometrically  squared  by  means  of  a  recti- 
linear figure  and  uniform  lines  of  the  same 
species  (equivalent  to  straights  as  to  compari- 
son inter  se) ;  but  an  F  form  circle  is  plani- 
fied  in  the  same  manner:  and  we  have  either 
the  Axiom  XI  of  Euclid  true  or  the  geomet- 
ric quadrature  of  the  circle,  although  thus 
far  it  has  remained  undecided,  which  of  these  ' 
two  has  place  in  reality. 

Whenever  tan2^  is  either  a  whole  number, 
or  a  rational  fraction,  whose  denominator  (re- 
duced to  the  simplest  form)  is  either  a  prime 
number  of  the  form  2m+l  (of  which  is  also 
2=2°+l),  or  a  product  of  however  many  prime 
numbers  of  this  form,  of  which  each  (with  the 


48        SCIENCE  ABSOLUTE  OP  SPACE. 

exception  of  2,  which  alone  may  occur  any 
number  of  times)  occurs  only  once  as  factor, 
we  can,  by  the  theory  of  polygons  of  the  illus- 
trious Gauss  (remarkable  invention  of  our, 
nay  of  every  age)  (and  only  for  such  values 
of  z) ,  construct  a  rectilineal  figure  =tan2^n  = 
area  Q5.  For  the  division  of  n  (the  theorem 
of  §  42  extending  easily  to  any  polygons)  mani- 
festly requires  the  partition  of  a  st.  Z,  which 
(as  can  be  shown)  can  be  achieved  geomet- 
rically only  under  the  said  condition. 

But  in  all  such  cases,  what  precedes  con- 
ducts easily  to  the  desired  end.  And  any  rec- 
tilineal figure  can  be  converted  geometrically 
into  a  regular  polygon  of  n  sides,  if  n  falls 
under  the  Gaussian  form. 

It  remains,  finally  (that  the  thing  may  be 
completed  in  every  respect),  to  demonstrate 
the  impossibility  (apart  from  any  supposition), 
of  deciding  a  priori,  whether  £,  or  some  S 
(and  which  one)  exists.  This,  however,  is  re- 
served for  a  more  suitable  occasion. 


APPENDIX  I. 


REMARKS   ON  THE  PRECEDING  TREATISE, 
BY   BOLYAI   PARKAS. 

[From  Vol.  II  of  Tentamen,  pp.  380-383.] 

Finally  it  may  be  permitted  to  add  something 
appertaining  to  the  author  of  the  Appendix  in 
the  first  volume,  who,  however,  may  pardon  me 
if  something  I  have  not  touched  with  his  acute- 
ness. 

The  thing  consists  briefly  in  this:  the  form- 
ulas of  spherical  trigonometry  (demonstrated 
in  the  said  Appendix  independently  of  Euclid's 
Axiom  XI)  coincide  with  the  formulas  of  plane 
trigonometry,  if  (in  a  way  provisionally  speak- 
ing) the  sides  of  a  spherical  triangle  are  ac- 
cepted as  reals,  but  of  a  rectilineal  triangle 
as  imaginaries;  so  that,  as  to  trigonometric 
formulas,  the  plane  may  be  considered  as  an 
imaginary  sphere,  if  for  real,  that  is  accepted 
in  which  sin  rt.  Z.  —  \. 

Doubtless,  of  the  Euclidean  axiom  has  been 
said  in  volume  first  enough  and  to  spare:  for 


50        SCIENCE  ABSOLUTE  OP  SPACE. 

the  case  if  it  were  not  true,  is  demonstrated 
(Tom.  I.  App.,  p.  13),  that  there  is  given  a  cer- 
tain /,  for  which  the  I  there  mentioned  is  =0 
(the  base  of  natural  logarithms),  and  for  this 
case  are  established  also  (ibidem,  p.  14)  the 
formulas  of  plane  trigonometry,  and  indeed  so, 
that  (by  the  side  of  p.  19,  ibidem)  the  formulas 
are  still  valid  for  the  case  of  the  verity  of  the 
said  axiom;  indeed  if  the  limits  of  the  values 
are  taken,  supposing  that  2=^=00;  truly  the 
Euclidean  system  is  as  if  the  limit  of  the  anti- 
Euclidean  (for  /=oo). 

Assume  for  the  case  of  i  existing,  the  unit 
=  ij  and  extend  the  concepts  sine  and  cosine 
also  to  imaginary  arcs,  so  that,  p  designating 
an  arc  whether  real  or  imaginary, 

!_     -±^    _  is  called  the 

2 
cosine  of  p,  and 

£_       ~e        _  is  called 


the  sine  of  p  (as  Tom.  L,  p.  177). 
Hence  for  q  real 


Q        -q 

e—  e        e 


—  1). 


SCIENCE  ABSOLUTE  OP  SPACE.        51 


q  _q  —  qsZl.NIIi          QNZ4.\^i 

0      e  -f  e        e  +e 

~~  ~~  =cos(—  ?v— 


if  of  course  also  in  the  imaginary  circle,  the 
sine  of  a  negative  arc  is  the  same  as  the  sine 
of  a  positive  arc  otherwise  equal  to  the  first, 
except  that  it  is  negative,  and  the  cosine  of  a 
positive  arc  and  of  a  negative  (if  otherwise 
they  be  equal)  the  same. 

In  the  said  Appendix,  §  25,  is  demonstrated 
absolutely,  that  is,  independently  of  the  said 
axiom;  that,  in  any  rectilineal  triangle  the 
sines  of  the  circles  are  as  the  circles  of  radii 
equal  to  the  sides  opposite. 

Moreover  is  demonstrated  for  the  case  of  i 
existing,  that  the  circle  of  radius  y  is 

=  ~i   i^__^  j  '  which,  for  /=!,  becomes 
*(«*_*-*). 

Therefore  (§31  ibidem],  for  a  right-angled 
rectilineal  triangle  of  which  the  sides  are  a 
and  b,  the  hypothenuse  c,  and  the  angles  oppo- 
site to  the  sides  a,  b,  c  are  «,  ,9,  rt.  Z,(for  /=!), 
in  I, 

lisin  a=*(^-^):*(^-O; 

and  so 

£c__0-c  e*—e~*         TTri  1 

l:sin«  =  —  -  --  :  --          Whence  1  :  sin  « 
2v_i  '  2v_i  ' 


52        SCIENCE  ABSOLUTE  OF  SPACE. 


=  —  sin    cV3):  —  sin    <zV__.     And  hence 


:  sn  «=sn    cl    :  sn    a*  _\. 

In  II  becomes 

cos  a  :  sin  /9=  cos  (aV^Tl)  :  1  ; 
in  III  becomes 

cos  (c\^Y)=cos  (<W^l).cos  (#V^i). 
These,  as  all  the  formulas  of  plane  trigonom- 
etry deducible  from  them,  coincide  completely 
with  the  formulas  of  spherical  trigonometry; 
except  that  if,  ex.  gr.,  also  the  sides  and  the 
angles  opposite  them  of  a  right-angled  spheri- 
cal triangle  and  the  hypothenuse  bear  the  same 
names,  the  sides  of  the  rectilineal  triangle  are 
to  be  divided  by  V—  1  to  obtain  the  formulas  for 
the  spherical  triangle. 

Obviously  we  get  (clearly  as  Tom.,  II.,  p.  252), 
from  I,  1  :  sin  «=sin  c  :  sin  a; 

from  II,  1  :  cos  a—  sin/5  :  cos  «; 

from  III,  cos  £=cos  a  cos  b. 

Though  it  be  allowable  to  pass  over  other 
things;  yet  I  have  learned  that  the  reader 
may  be  offended  and  impeded  by  the  deduc- 
tion omitted,  (Tom.  I.,  App.,  p.  19)  [in  §  32  at 
end]  :  it  will  not  be  irrelevant  to  show  how,  ex. 
gr.,  from 


follows 


SCIENCE  ABSOLUTE  OF  SPACE.        53 


(the  theorem  of  Pythagoras  for  the  Euclidean 
system)  ;  probably  thus  also  the  author  de- 
duced it,  and  the  others  also  follow  in  the 
same  manner. 

Obviously  we  have,  the  powers  of  e  being  ex- 
pressed by  series  (like  Tom.  L,  p.  168), 


2.3./3f2.3.4.*4 

/-4 

and  so 


2.3.*82.3.4.* 


1  3.4.*4'3.4.5.6.*6 
=2H — i^,  (designating  by 

-  the  sum  of  all  the  terms  after—,  !  ;    and  we 

i~  i>'~) 

have  ?/=0,    while   /^=oc.     For  all  the    terms 
which    follow  —   are  divided  by  i*;  the  first 

term  will  be  — — 2;   and  any  ratio    <-^;    and 

though  the   ratio  everywhere  should    remain 
this,  the  sum  would  be  ^Tom.  I.,  p.  131), 

/v  I   -i        A/  §C 


which  manifestly  =?=0,  while  /= 
And  from 


54        SCIENCE  ABSOLUTE  OP  SPACE. 


f 

^ 


(a+b)         -(a+b)         a-b         -(a-b)  ") 

\     +e      i     +e  i   +e     i    J 
follows  (for  w,  z;,  A  taken  like  ^) 


And  hence 

C2=- 


which  =a2+l>2. 


APPENDIX  II. 


SOME    POINTS    IN    JOHN    BOLYAl's    APPENDIX 

COMPARED   WITH   LOBACHEVSKI, 

BY    WOLFGANG    BOLYAI. 

[From  Kurzer  Grundriss,  p.  82.] 

Lobachevski  and  the  author  of  the  Appendix 
each  consider  two  points  A,  B,  of  the  sphere- 
limit,  and  the  corresponding  axes 
ray  AM,  ray  BN  (§  23). 

They  demonstrate  that,  if  «,  ,?, 
r  designate  the  arcs  of  the  circle 
limit  AB,  CD,  HL,  separated  by 
r  segments  of  the  axis  AC=1,  AH 
—x,  we  have 

Mi).' 

Lobachevski   represents  the  value  of    -  by 

0~x,  e  having  some  value  >1,  dependent  on  the 
unit  for  length  that  we  have  chosen,  and  able 
to  be  supposed  equal  to  the  Naperian  base. 

The  author  of  the  Appendix  is  led  directly 
to  introduce  the  base  of  natural  logarithms. 


56        SCIENCE  ABSOLUTE  OP  SPACE. 

If  we  put  ^=d,  and  r,  rr  are  arcs  situated  at 
the  distances  y,  i  from  «,  we  shall  have 
-=ay=Y,         -,=^=1,  whence   Y=IT 

He  demonstrates  afterward  (§  29)  that,  if  u 
is  the  angle  which  a  straight  makes  with  the 
perpendicular  y  to  its  parallel,  we  have 
Y=cot  \u. 

Therefore,  if  we  put  z=-~—u,  we  have 
Y=tan 


1—  tan  z  tan  j« 

whence  we  get,  having  regard  to  the  value  of 
tan  J«=Y~1, 

tan  z^  (Y-Y^NiJl'-J    ']  (§30). 
If  now  jv  is  the   semi-chord  of  the   arc  of 

/^ 

circle-limit   2^,   we    prove   (§  30)   that  - 

tan  z 

constant. 

Representing  this  constant  by  i,  and  making 
y  tend  toward  zero,  we  have 

—  =1,  whence 

2r 

2y 

T   '  _1 

2=2  *  tan  z=i  -  -—  i 


SCIENCE  ABSOLUTE  OP  SPACE.       57 
or  putting  -^-=k,  \—el, 


/  being-  infinitesimal  at  the  same  time  as  k. 
Therefore,  for  the  limit,  1  =  1  and  consequently 
I=& 

The  circle  traced  on  the  sphere-limit  with 
the  arc  r  of  the  curve-limit  for  radius,  has  for 
length  2-r.  Therefore, 

OjK=2rr=2-*  tan  z=*i  (Y— Y"1). 

In  the  rectilineal  A  where  a,  p  designate  the 
angles  opposite  the  sides  a,  b,  we  have  (§  25) 

sin  a:sin  ,i=Qa:Qt>=-i(A—A-1):  -i(B— B"1) 
=sin  (^^l)  :sin  (fc— I). 

Thus  in  plane  trigonometry  as  in  spherical 
trigonometry,  the  sines  of  the  angles  are  to 
each  other  as  the  sines  of  the  opposite  sides, 
only  that  on  the  sphere  the  sides  are  reals, 
and  in  the  plane  we  must  consider  them  as 
imaginaries,  just  as  if  the  plane  were  an 
imaginary  sphere. 

We  may  arrive  at  this  proposition  without  a 
preceding  determination  of  the  value  of  I. 

0* 

If  we  designate  the  constant by  q,  we 

tan  z 

shall  have,  as  before 

=*q  (Y-Y-1), 


58        SCIENCE  ABSOLUTE  OF  SPACE. 

whence  we  deduce  the  same  proportion  as 
above,  taking  for  i  the  distance  for  which  the 
ratio  I  is  equal  to  e. 

If  axiom  XI  is  not  true,  there  exists  a  de- 
terminate i,  which  must  be  substituted  in  the 
formulas. 

If,  on  the  contrary,  this  axiom  is  true,  we 
must  make  in  the  formulas  i=  oo.  Because,  in 

this  case,  the  quantity  -=Y  is  always  =1,  the 

sphere-limit  being  a  plane,  and  the  axes  being 
parallel  in  Euclid's  sense. 

The  exponent  \  must  therefore  be  zero,  and 
consequently  i=  GO. 

It  is  easy  to  see  that  Bolyai's  formulas  of 
plane  trigonometry  are  in  accord  with  those  of 
Lobachevski. 

Take  for  example  the  formula  of  §  37, 

tan  //  (#)=sin  B  tan  //(/), 

a  being  the  hypothenuse  of  a  right-angled  tri- 
angle, p  one  side  of  the  right  angle,  and  B  the 
angle  opposite  to  this  side. 

Bolyai's  formula  of  §  31,  I,  gives 

1  :  sin  B=(A-A-1):(P-P-]). 

Now,    putting    for   brevity,    i/7  (Jc)=k' ,    we 
have  tan  2p ' :  tan  2a '  —  (cot  a '  — tan  a ' }  :  (cot  p ' 
/  =  A-A-1       P-P-1^!  :  sin  B. 


APPENDIX  III. 


LIGHT  FROM  NON-EUCLIDEAN  SPACES  ON  THE 
TEACHING   OF    ELEMENTARY  GEOMETRY. 

BY  G.   B.   HALSTED. 

As  foreshadowed  by  Bolyai  and  Riemann, 
founded  by  Cayley,  extended  and  interpreted 
for  hyperbolic,  parabolic,  elliptic  spaces  by 
Klein,  recast  and  applied  to  mechanics  by  Sir 
Robert  Ball,  projective  metrics  may  be  looked 
upon  as  characteristic  of  what  is  highest  and 
most  peculiarly  modern  in  all  the  bewildering 
range  of  mathematical  achievement. 

Mathematicians  hold  that  number  is  wholly 
a  creation  of  the  human  intellect,  while  on  the 
contrary  our  space  has  an  empirical  element. 
Of  possible  geometries  we  can  not  say  a  priori 
which  shall  be  that  of  our  actual  space,  the 
space  in  which  we  move.  Of  course  an  ad- 
vance so  important,  not  only  for  mathemat- 
ics but  for  philosophy,  has  had  some  m^taphy- 
sical  opponents,  and  as  long  ago  as  1878  I 
mentioned  in  my  Bibliography  of  Hyper- 


60        SCIENCE  ABSOLUTE  OP  SPACE. 


Space  and  Non-Euclidean  Geometry  (American 
Journal  of  Mathematics,  Vol.  I,  1878,  Vol.  II, 
1879)  one  of  these,  Schmitz-Dumont,  as  a  sad 
paradoxer,  and  another,  J.  C.  Becker,  both  of 
whom  would  ere  this  have  shared  the  oblivion 
of  still  more  antiquated  fighters  against  the 
light,  but  that  Dr.  Schotten,  praiseworthy  for 
the  very  attempt  at  a  comparative  planimetry, 
happens  to  be  himself  a  believer  in  the  a  priori 
founding  of  geometry,  while  his  American  re- 
viewer, Mr.  Ziwet,  was  then  also  an  anti-non- 
Euclidean,  though  since  converted. 

He  says,  ' '  we  find  that  some  of  the  best  Ger- 
man text  books  do  not  try  at  all  to  define  what 
is  space,  or  what  is  a  point,  or  even  what  is  a 
straight  line."  Do  any  German  geometries  de- 
fine space?  I  never  remember  to  have  met  one 
that  does. 

In  experience,  what  comes  first  is  a  bounded 
surface,  with  its  boundaries,  lines,  and  their 
boundaries,  points.  Are  the  points  whose 
definitions  are  omitted  anything  different  or 
better? 

Dr.  Schotten  regards  the  two  ideas  "direc- 
tion" and  "distance"  as  intuitively  given  in 
the  mind  and  as  so  simple  as  to  not  require 
definition. 

When    we    read   of    two   jockeys   speeding 


SCIENCE  ABSOLUTE  OF  SPACE.        61 

around  a  track  in  opposite  directions,  and 
also  on  page  87  of  Richardson's  Euclid,  1891, 
read,  ' '  The  sides  of  the  figure  must  be  pro- 
duced in  the  same  direction  of  rotation ;  .  .  . 
going  round  the  figure  always  in  the  same 
direction,"  we  do  not  wonder  that  when  Mr. 
Ziwet  had  written:  "he  therefore  bases  the 
definition  of  the  straight  line  on  these  two 
ideas,"  he  stops,  modifies,  and  rubs  that  out 
as  follows,  "or  rather  recommends  to  eluci- 
date the  intuitive  idea  of  the  straight  line 
possessed  by  any  well-balanced  mind  by  means 
of  the  still  simpler  ideas  of  direction"  [in  a 
circle]  "and  distance"  [on  a  curve]. 

But  when  we  come  to  geometry  as  a  science, 
as  foundation  for  work  like  that  of  Cayley  and 
Ball,  I  think  with  Professor  Chrystal:  "  It  is 
essential  to  be  careful  with  our  definition  of  a 
straight  line,  for  it  will  be  found  that  vir- 
tually the  properties  of  the  straight  line  de- 
termine the  nature  of  space. 

* '  Our  definition  shall  be  that  two  points  in 
general  determine  a  straight  line." 

We  presume  that  Mr.  Ziwet  glories  in  that 
unfortunate  expression  "a  straight  line  is  the 
shortest  distance  between  two  points,"  still 
occurring  in  Wentworth  (New  Plane  Geom- 
etry, page  33),  even  after  he  has  said,  page  5, 


62        SCIENCE  ABSOLUTE  OF  SPACE. 

"the  length  of  the  straight  line  is  called  the 
distance  between  two  points."  If  the  length 
of  the  one  straight  line  between  two  points  is 
the  distance  between  those  points,  how  can  the 
straight  line  itself  be  the  shortest  distance? 
If  there  is  only  one  distance,  it  is  the  longest 
as  much  as  the  shortest  distance,  and  if  it  is 
the  length  of  this  shorto-longest  distance 
which  is  the  distance  then  it  is  not  the 
straight  line  itself  which  is  the  longo-shortest 
distance.  But  Wentworth  also  says:  "Of  all 
lines  joining  two  points  the  shortest  is  the 
straight  line." 

This  general  comparison  involves  the  meas- 
urement of  curves,  which  involves  the  theory 
of  limits,  to  say  nothing  of  ratio.  The  very 
ascription  of  length  to  a  curve  involves  the 
idea  of  a  limit.  And  then  to  introduce  this 
general  axiom,  as  does  Wentworth,  only  to 
prove  a  very  special  case  of  itself,  that  two 
sides  of  a  triangle  are  together  greater  than 
the  third,  is  surely  bad  logic,  bad  pedagogy, 
bad  mathematics. 

This  latter  theorem,  according  to  the  first 
of  Pascal's  rules  for  demonstrations,  should 
not  be  proved  at  all,  since  every  dog  knows  it. 
But  to  this  objection,  as  old  as  the  sophists, 
Simson  long  ago  answered  for  the  science  of 


SCIENCE  ABSOLUTE  OF  SPACE.        63 

geometry,  that  the  number  of  assumptions 
ought  not  to  be  increased  without  necessity  ; 
or  as  Dedekind  has  it:  "  Was  beweisbar  ist, 
soil  in  der  Wissenschaft  nicht  ohne  Beuueis 
geglaubt  werden" 

Professor  W.  B.  Smith  (Ph.  D.,  Goettingen), 
has  written:  "  Nothing  could  be  more  unfor- 
tunate than  the  attempt  to  lay  the  notion  of 
Direction  at  the  bottom  of  Geometry. ' ' 

Was  it  not  this  notion  which  led  so  good  a 
mathematician  as  John  Casey  to  give  as  a 
demonstration  of  a  triangle's  angle-sum  the 
procedure  called  "  a  practical  demonstration  " 
on  page  87  of  Richardson's  Euclid,  and  there 
described  as  * '  laying  a  *  straight  edge  '  along 
one  of  the  sides  of  the  figure,  and  then  turn- 
ing it  round  so  as  to  coincide  with  each  side  in 
turn." 

This  assumes  that  a  segment  of  a  straight 
line,  a  sect,  may  be  translated  without  rota- 
tion, which  assumption  readily  comes  to  view 
when  you  try  the  procedure  in  two-dimensional 
spherics.  Though  this  fallacy  was  exposed  by 
so  eminent  a  geometer  as  Olaus  Henrici  in  so 
public  a  place  as  the  pages  of  'Nature,'  yet  it 
has  just  been  solemnly  reproduced  by  Pro- 
fessor G.  C.  Edwards,  of  the  University  of 
California,  in  his  Elements  of  Geometry:  Mac- 


64        SCIENCE  ABSOLUTE  OF  SPACE. 

Millan,  1895.  It  is  of  the  greatest  importance 
for  every  teacher  to  know  and  connect  the 
commonest  forms  of  assumption  equivalent  to 
Euclid's  Axiom  XL  If  in  a  plane  two  straight 
lines  perpendicular  to  a  third  nowhere  meet, 
are  there  others,  not  both  perpendicular  to 
any  third,  which  nowhere  meet?  Euclid's 
Axiom  XI  is  the  assumption  No.  Playf air's 
answers  no  more  simply.  But  the  very  same 
answer  is  given  by  the  common  assumption  of 
our  geometries,  usually  unnoticed,  that  a  circle 
may  be  passed  through  any  three  points  not 
costraight. 

This  equivalence  was  pointed  out  by  Bolyai 
Farkas,  who  looks  upon  this  as  the  simplest 
form  of  the  assumption.  Other  equivalents 
are,  the  existence  of  any  finite  triangle  whose 
angle-sum  is  a  straight  angle;  or  the  existence 
of  a  plane  rectangle;  or  that,  in  triangles,  the 
angle-sum  is  constant. 

One  of  Legendre's  forms  was  that  through 
every  point  within  an  angle  a  straight  line 
may  be  drawn  which  cuts  both  arms. 

But  Legendre  never  saw  through  this  mat- 
ter because  he  had  not,  as  we  have,  the  eyes 
of  Bolyai  and  Lobachevski  to  see  with.  The 
same  lack  of  their  eyes  has  caused  the  author 
of  the  charming  book  "  Euclid  and  His  Modern 


SCIENCE  ABSOLUTE  OF  SPACE.        65 

Rivals,"  to  give  us  one  more  equivalent  form: 
"In  any  circle,  the  inscribed  equilateral  tetra- 
gon is  greater  than  any  one  of  the  segments 
which  lie  outside  it."  (A  New  Theory  of 
Parallels  by  C.  L.  Dodgson,  3d.  Ed.,  1890.) 

Any  attempt  to  define  a  straight  line  by 
means  of  "direction"  is  simply  a  case  of  "ar- 
gumentum  in  circulo."  In  all  such  attempts 
the  loose  word  "direction"  is  used  in  a  sense 
which  presupposes  the  straight  line.  The 
directions  from  a  point  in  Euclidean  space  are 
only  the  oc2  rays  from  that  point. 

Rays  not  costraight  can  be  said  to  have  the 
same  direction  only  after  a  theory  of  parallels 
is  presupposed,  assumed. 

Three  of  the  exposures  of  Professor  G.  C. 
Edwards'  fallacy  are  here  reproduced.  The 
first,  already  referred  to,  is  from  Nature,  Vol. 
XXIX,  p.  453,  March  13,  1884. 

"I  select  for  discussion  the  'quaternion 
proof  "  given  by  Sir  William  Hamilton.  .  .  . 
Hamilton's  proof  consists  in  the  following: 

"One  side  AB  of  the  triangle  ABC  is  turned 
about  the  point  B  till  it  lies  in  the  continuation 
of  BC;  next,  the  line  BC  is  made  to  slide  along 
BC  till  B  comes  to  C,  and  is  then  turned  about 
C  till  it  comes  to  lie  in  the  continuation  of  AC. 


66        SCIENCE  ABSOLUTE  OF  SPACE. 

*  *  It  is  now  again  made  to  slide  along  CA  till 
the  point  B  comes  to  A,  and  is  turned  about  A 
till  it  lies  in  the  line  AB.  Hence  it  follows, 
since  rotation  is  independent  of  translation, 
that  the  line  has  performed  a  whole  revolution, 
that  is,  it  has  been  turned  through  four  right 
angles.  But  it  has  also  described  in  succession 
the  three  exterior  angles  of  the  triangle,  hence 
these  are  together  equal  to  four  right  angles, 
and  from  this  follows  at  once  that  the  interior 
angles  are  equal  to  two  right  angles. 

' '  To  show  how  erroneous  this  reasoning  is— 
in  spite  of  Sir  William  Hamilton  and  in  spite 
of  quaternions — I  need  only  point  out  that  it 
holds  exactly  in  the  same  manner  for  a  triangle 
on  the  surface  of  the  sphere,  from  which  it 
would  follow  that  the  sum  of  the  angles  in  a 
spherical  triangle  equals  two  right  angles, 
whilst  this  sum  is  known  to  be  always  greater 
than  two  right  angles.  The  proof  depends 
only  on  the  fact,  that  any  line  can  be  made  to 
coincide  with  any  other  line,  that  two  lines  do 
so  coincide  when  they  have  two  points  in  com- 
mon, and  further,  that  a  line  may  be  turned 
about  any  point  in  it  without  leaving  the  sur- 
face. But  if  instead  of  the  plane  we  take  a 
spherical  surface,  and  instead  of  a  line  a  great 


SCIENCE  ABSOLUTE  OP  SPACE.        67 

circle  on  the  sphere,  all  these  conditions  are 
again  satisfied. 

4  *  The  reasoning  employed  must  therefore 
be  fallacious,  and  the  error  lies  in  the  words 
printed  in  italics;  for  these  words  contain  an 
assumption  which  has  not  been  proved. 

"O.  HENRICI." 

Perronet  Thompson,  of  Queen's  College, 
Cambridge,  in  a  book  of  which  the  third  edi- 
tion is  dated  1830,  says: 

'''Professor  Playfair,  in  the  Notes  to  his 
'Elements  of  Geometry'  [1813],  has  proposed 
another  demonstration,  founded  on  a  remark- 
able non  causa  pro  causa. 

"It  purports  to  collect  the  fact  [Eu.  I.,  32, 
Cor.,  2]  that  (on  the  sides  being  successively 
prolonged  to  the  same  hand)  the  exterior 
angles  of  a  rectilinear  triangle  are  together 
equal  to  four  right  angles,  from  the  circum- 
stance that  a  straight  line  carried  round  the 
perimeter  of  a  triangle  by  being  applied  to  all 
the  sides  in  succession,  is  brought  into  its  old 
situation  again;  the  argument  being,  that  be- 
cause this  line  has  made  the  sort  of  somerset 
it  would  do  by  being  turned  through  four 
right  angles  about  a  fixed  point,  the  exterior 


68 


SCIENCE  ABSOLUTE  OF  SPACE. 


angles  of  the  triangle  have  necessarily  been 
equal  to  four  right  angles. 

"The  answer  to  which  is,  that  there  is  no 
connexion  between  the  things  at  all,  and  that 
the  result  will  just  as  much  take  place  where 
the  exterior  angles  are  avowedly  not  equal  to 
four  right  angles. 

*  'Take,  for  example,  the  plane  triangle  formed 
by  three  small  arcs  of  the  same  or  equal  circles, 

as  in  the  margin; 
and  it  is  manifest 
that  an  arc  of  this 
circle  may  be  car- 
ried  round  pre- 
cisely in  the  way 
described  and  re- 
turn to  its  old  sit- 
uation,  and  yet 
there  be  no  pre- 
tense for  infer- 
ring that  the  exterior  angles  were  equal  to 
four  right  angles. 

"And  if  it  is  urged  that  these  are  curved 
lines  and  the  statement  made  was  of  straight; 
then  the  answer  is  by  demanding  to  know, 
what  property  of  straight  lines  has  been  laid 
down  or  established,  which  determines  that 
what  is  not  true  in  the  case  of  other  lines  is 


SCIENCE  ABSOLUTE  OP  SPACE.       69 

true  in  theirs.  It  has  been  shown  that,  as  a 
general  proposition,  the  connexion  between  a 
line  returning  to  its  place  and  the  exterior 
angles  having  been  equal  to  four  right  angles, 
is  a  non  sequitur  ;  that  it  is  a  thing  that  may 
be  or  may  not  be;  that  the  notion  that  it  re- 
turns to  its  place  because  the  exterior  angles 
have  been  equal  to  four  right  angles,  is  a  mis- 
take. From  which  it  is  a  legitimate  conclu- 
sion, that  if  it  had  pleased  nature  to  make  the 
exterior  angles  of  a  triangle  greater  or  less 
than  four  right  angles,  this  would  not  have 
created  the  smallest  impediment  to  the  line's 
returning  to  its  old  situation  after  being  car- 
ried round  the  sides;  and  consequently  the 
line's  returning  is  no  evidence  of  the  angles 
not  being  greater  or  less  than  four  right 
angles." 

Charles  L.  Dodgson,  of  Christ  Church,  Ox- 
ford, in  his  "  Curiosa  Mathematica,"  Part  I, 
pp.  70-71,  3d  Ed.,  1890,  says: 

"Yet  another  process  has  been  invented— 
quite  fascinating  in  its  brevity  and  its  ele- 
gance— which,  though  involving  the  same  fal- 
lacy as  the  Direction-Theory,  proves  Euc.  I, 
32,  without  even  mentioning  the  dangerous 
word  'Direction.' 


70        SCIENCE  ABSOLUTE  OP  SPACE. 

1 '  We  are  told  to  take 
any  triangle  ABC;  to 
produce  CA  to  D;  to 
make  part  of  CD,  viz., 
AD,  revolve,  about  A, 
into  the  position  ABE; 
then  to  make  part  of  this 
line,  viz.,  BE,  revolve, 
about  B,  into  the  position  BCF;  and  lastly  to 
make  part  of  this  line,  viz.,  CF,  revolve,  about 
C,  till  it  lies  along  CD,  of  which  it  originally 
formed  a  part.  We  are  then  assured  that  it 
must  have  revolved  through  four  right  angles: 
from  which  it  easily  follows  that  the  interior 
angles  of  the  triangle  are  together  equal  to 
two  right  angles. 

' '  The  disproof  of  this  fallacy  is  almost  as 
brief  and  elegant  as  the  fallacy  itself.  We 
first  quote  the  general  principle  that  we  can 
not  reasonably  be  told  to  make  a  line  fulfill 
two  conditions,  either  of  which  is  enough  by 
itself  to  fix  its  position:  e.  g.,  given  three 
points  X,  Y,  Z,  we  can  not  reasonably  be  told 
to  draw  a  line  from  X  which  shall  pass 
through  Y  and  Z:  we  can  make  it  pass 
through  Y,  but  it  must  then  take  its  chance 
of  passing  through  Z;  and  vice  versa. 

"Now  let  us  suppose  that,  while  one  part  of 


SCIENCE  ABSOLUTE  OP  SPACE.        71 

AE,  viz.,  BE,  revolves  into  the  position  BF, 
another  little  bit  of  it,  viz.,  AG,  revolves, 
through  an  equal  angle,  into  the  position  AH; 
and  that,  while  CF  revolves  into  the  position 
of  lying  along  CD,  AH  revolves — and  here 
comes  the  fallacy. 

"You  must  not  say  *  revolves,  through  an 
equal  angle,  into  the  position  of  lying  along 
AD, '  for  this  would  be  to  make  AH  fulfill  two 
conditions  at  once. 

' '  If  you  say  that  the  one  condition  involves 
the  other,  you  are  virtually  asserting  that  the 
lines  CF,  AH  are  equally  inclined  to  CD — and 
this  in  consequence  of  AH  having  been  so 
drawn  that  these  same  lines  are  equally  in- 
clined to  AE. 

"That  is,  you  are  asserting,  'A  pair  of  lines 
which  are  equally  inclined  to  a  certain  trans- 
versal, are  so  to  any  transversal. '  [Deducible 
fromEuc.  I,  27,  28,  29.]" 


MATHEMATICAL   WORKS 

BY 

GEORGE   BRUCE   HALSTED, 

A.M.  (PRINCETON);  PH.  D.  (JOHNS  HOPKINS);  EX-FELLOW  OF  PRINCETON 
COLLEGE;  TWICE  FELLOW  OF  JOHNS  HOPKINS  UNIVERSITY;  INTERCOL 
LEGIATK    PRIZEMAN;     SOMETIME    INSTRUCTOR   IN    POST    GRADUATE 
MATHEMATICS,  PRINCETON  COLLEGE;  PROFESSOR  OF  MATHEMAT- 
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AMERICAN  MATHEMATICAL  SOCIETY;  MEMBER  OF  THK  LON- 
DON MATHEMATICAL  SOCIETY;    MEMBER  OF  THE  ASSO- 
CIATION FOR  THE  IMPROVEMENT  OF  GEOMETRICAL 
TEACHING;   EHRENMITGLIED   DES   COMITES   DES 
LOBACHEVSKY-CAPITALS  ;  MIEMBRO  DE  LA  So- 

CIEDAD  ClENTIFICA  "ALZATK"  DE  MEXICO', 
SOCIO    CORRESPONSAL  DE   LA  SOCIEDAD  DE   GEOGRAFIA  Y   ESTADISTICO 

DE  MEXICO;  SOCIETAIRE   PERPETUAL  DE  LA  SOCIETE  MATHE- 

MATIQUE   DE    FRANCE;    SOCIO    PERPETUO  DELLA   ClBCOLO 

MATEMATICO    DI    PALERMO;     PBESIDENT   OF 
THE  TEXAS  ACADEMY  OF  SCIENCE. 

Mensuration.     4th  Ed.     1892.    $1.10. 
Ginn  &  Co.     Boston.  U.  S.  A.,  and  London. 

Elements  of  Geometry.    6th  Ed.    1893.    $1.75. 
John  Wiley  &  Sons.    53  E.  10th  St.,  New  York. 
Chapman  &  Hall.     London. 

Synthetic  Geometry.    2nd  Ed.    1893.     $1.50. 
John  Wiley  &  Sons.    53  E.  10th  St.,  New  York. 

Lobachevski's  Non-Euclidean  Geometry.    4th  Ed.    1891.     $1. 
G.  B.  Halsted.  2407  Guadalupe  St.,  Austin,  Texas,  U.  S.  A. 

Bolyai's  Science  Absolute  of  Space.    4th  Ed.    1896.     $1.00. 
G.  B.  Halsted,  2407  Guadalupe  St.,  Austin,  Texas,  U.  S.  A. 

Vasiliev  on  Lobachevski.    1894.    50c. 

G.  B.  Halsted,  2407  Guadalupe  St.,  Austin,  Texas,  U.  S.  A. 


Sent  postpaid  on  receipt  of  the  price. 


VOLUME  ONE  OF  THE  NEOMONIC   SERIES. 

NICOLAI  IVANOVICH  LOBACHEVSKI. 

BY  A.  VASIL.IEV. 

Translated  from  the  Russian  by 

GEORGE  BRUCE  HALSTED. 

From  a  six-column  Review  of  this  Translation  in  Science,  March 

29, 1895: 

"Non-Euclidian  Geometry,  a  subject  which  has  not  only  revo- 
lutionized geometrical  science,  but  has  attracted  the  attention  of 
physicists,  psychologists  and  philosophers." 

"Without  question  the  best  and  most  authentic  source  of  infor- 
mation on  this  original  thinker." 


From  a  two-column  Review  in  the  Nation,  April  4,  1895,  by  C.  8. 

Pierce: 

"Kazfin  was  not  the  milieu  for  a  man  of  genius,  especially  not 
for  so  profound  a  genius  as  that  of  Lobachevski." 

"All  of  Lobachevski's  writings  are  marked  by  the  same  high- 
strung  logic." 


I  have  read  it  with  intense  interest.  By  issuing  this  transla- 
tion you  have  put  American  readers  under  renewed  obligation  to 
you.  FLORIAN  CAJORI. 


I  have  read  with  great  interest  your  translation  of  the  address 
in  commemoration  of  Lobachevski.  It  is  a  most  fortunate  thing 
for  us  in  the  rank  and  file  that  you  have  maintained  such  an  in- 
terest in  the  history  of  this  non-Euclidian  work;  for  while  you 
have  conquered  for  Saccheri,  Bolyai  and  the  rest  the  share  of 
fame  that  is  their  due,  you  have  made  it  impossible  for  American 
teachers  of  any  spirit  to  shut  their  eyes  to  the  "  hypothesis  anguli 
acuti."  Very  truly  yours, 

G.  H.  LOUD, 
Professor  of  Mathematics  in  Colorado  College. 


BURLINGTON,  VT.,  October  19th,  1894. 
I  am  astonished  to  find  these  researches  of  such  deep,,  philo- 


sophical  import.    You  many  congratulate  yourself  on  your  in- 
strumentality in  spreading  the  news  in  America. 

Very  sincerely,  A.  L.  DANIELS, 

Professor  of  Mathematics,  University  of  Vermont. 


STAUNTON.  VA.,  October  13th,  1894. 

The  history  of  the  life  and  work  of  such  a  man  as  Lobachevski 
will  be  a  grand  inspiration  to  mathematicians,  especially  with 
such  a  leader  as  yourself,  in  the  important  field  of  non-Euclidean 
Geometry.  Very  truly  yours.  G.  B.  M.  ZERR. 


BETHLEHEM.  PA..  October  22nd,  1894. 

I  have  read  the  Lobachevski  with  much  pleasure  and — what  is 
better — profit.        Yours  very  truly.  G.  L.  DOOLITTLE, 

Professor  of  Mathematics  in  the  University  of  Pennsj'lvania. 


HALLE  a.  S.,  LAFONTAINESTR.,  2;  23,10,  '94. 
Hochgeehrter  Herr: 

Auf  der  Naturforscherversamrnlung  in  Wien  lernte  ich  Prof. 
Wasilief  ans  Kas;ln  kennen,  der  mir  erzaehlte.  das  Sie  seine  Kede 
bei  der  Lobatschefsky-Feier  uebersetzen  wollten.  Diese  Xach- 
richt  war  mich  sehr  willkommen.  da  die  russische  mir  unver- 
standlich  ist. 

Nun  erhalte  ich  heute  von  Ihnen  diese  Uebersetzungzugesandt 
und  sage  Ihnen  dafuer  meinen  verbindlichsten  Dank.  Sie  haben 
mit  der  Uebersetzung  dieser  interessauten  Rede  sich  den  Ans- 
pruch  auf  den  Dank  der  mathematischen  Welt  erworben  ! 

Hochachtungsvoll  Ihr  ergebener,  STAECKEL. 


STANFORD  UNIVERSITY, 
PALO  ALTO,  CAL.,  October  19th,  1894. 

1  have  read  the  Lobachevski  with  the  greatest  interest,  and  re- 
joice that  you.  "in  the  midst  of  the  virgin  forests  of  Texas."  are 
able  to  do  this  work.  And.  by  the  way,  I  have  heard  at  different 
times  a  number  of  professors  speak  of  your  Geometry  (Elements). 
All  who  have  examined  it.  and  whom  I  have  heard  speak  of  it, 
seem  to  think  it  the  best  Geometry  we  have. 

Yours  truly.  A.  P.  CARMAN, 

Professor  of  Physics,  Leland  Stanford.  Jr..  University. 


READY  FOR  THE  PRESS. 

VOLUME   FOUR  OF  THE   NEOMONIC   SERIES. 

THE  LIFE  OF  BOLYAI. 

From  Hungarian  (Magyar)  sources,  by  Dr.  George  Bruce 
Halsted.  [Containing  the  Autobiography  of  Bolyai  Farkas,  now 
first  translated  from  the  Magyar.] 


VOLUME  FIVE  OF  THE  NEOMONIC  SERIES. 

NEW  ELEMENTS  OF  GEOMETRY 

WITH  A   COMPLETE   THEORY  OF   PARALLELS. 
BY  N.  I.  LOBACHEVSKI. 

Translated  from  the  Russian  by 
DR.  GEORGE  BRUCE  HALSTED. 

[Though  this  is  Lobachevski's  greatest  work,  it  has  never  be- 
fore been  translated  out  of  the  Russian  into  any  other  language 
whatever.] 


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